Mixed Frequency Models
Summary and Keywords
The majority of econometric models ignore the fact that many economic time series are sampled at different frequencies. A burgeoning literature pertains to econometric methods explicitly designed to handle data sampled at different frequencies. Broadly speaking these methods fall into two categories: (a) parameter driven, typically involving a state space representation, and (b) data driven, usually based on a mixed-data sampling (MIDAS)-type regression setting or related methods. The realm of applications of the class of mixed frequency models includes nowcasting—which is defined as the prediction of the present—as well as forecasting—typically the very near future—taking advantage of mixed frequency data structures. For multiple horizon forecasting, the topic of MIDAS regressions also relates to research regarding direct versus iterated forecasting.
Most econometric models ignore the fact that many economic time series are sampled at different frequencies. Indeed, the most convenient approach is to aggregate high-frequency data to obtain a balanced data set at the same low frequency.
This practice has at least two undesirable consequences: (a) potential distortions of dynamic relationships among economic variables and (b) forgoing the possibility to take advantage of the real-time flow of economic data releases.
For example, if a researcher has quarterly data on Gross Domestic Product (GDP) growth with monthly data on Industrial Production (IP), and wants to study the relationship between GDP and IP, common practice is to aggregate the IP data to the quarterly frequency, by either taking an average of the monthly data or using the last month of each quarter. While simple, in general temporal aggregation entails a loss of information. This implies that key econometric features, such as Granger causality or exogeneity, can be spuriously modified when working with aggregated data (see, e.g., Granger, 1980; Marcellino, 1999; Ghysels, Hill, & Motegi, 2015, 2016 for details).
Decision makers, and policymakers in particular, also need to assess in real time the current state of the economy and its expected developments. GDP is released quarterly (and with a substantial temporal delay), while a range of leading and coincident indicators is timely available at a monthly or even higher frequency. Hence, we may want to construct a forecast of the current quarter GDP growth based on the available higher frequency information. Mixed frequency data in a forecasting setting invariably relate to the notion of nowcasting.
A burgeoning literature pertains to econometric methods explicitly designed to handle data sampled at different frequencies. The realm of applications of the class of mixed frequency models includes nowcasting—which is defined as the prediction of the present—as well as forecasting—typically the very near future—taking advantage of mixed frequency data structures. For multiple horizon forecasting, the topic of mixed-data sampling (MIDAS) regressions also relates to research regarding direct versus iterated forecasting.
Continuing with GDP forecasting as an example, nowcasting in this context means that in a particular calendar month, GDP for the current quarter is not observed. It can even be the case that GDP is available only with considerable publication delay beyond the end of the quarter. The question then arises whether we can make a prediction about the current quarter GDP using monthly, weekly, daily economic time series.
Broadly speaking these methods fall into two categories: (a) settings involving a state space representation and (b) mixed-data sampling (MIDAS) type regressions or related methods. State space models involve latent processes, and therefore rely on filtering to extract hidden states that are used in order to predict future outcomes. State space models are, using the terminology of Cox (1981), parameter-driven models. The MIDAS regression models are, using again the same terminology, observation-driven models, as they are formulated exclusively in terms of observable data.
State space models involving mixed frequency data have a long history. Early contributions include Harvey and Pierse (1984), Bernanke, Gertler, and Watson (1997), Zadrozny (1990), Mariano and Murasawa (2003), and Mittnik and Zadrozny (2004). More recent contributions include Nunes (2005), Giannone, Reichlin, and Small (2008), Aruoba, Diebold, and Scotti (2009), Ghysels and Wright (2009), Marcellino and Schumacher (2010), Foroni and Marcellino (2014), and Schorfheide and Song (2015); Eraker, Chiu, Foerster, Kim, and Seoane (2015), among others, have further explored state space models in a mixed frequency setting.
Two illustrative examples of state space models in a mixed frequency data setting are considered. First, Zadrozny (1990) looks at a state space representation of ARMA models, and studies a bivariate monthly model of unemployment and GDP growth:
where , and , , . The matrices depend on a parameter vector Moreover, is positive definite. is , is lower triangular, and is Partition of stocks and flows results in:
with . Let be a vector of “potential” observations on ,
Abstracting from observation errors, we have:
where are diagonal indicator matrices with zeros and ones. Note that can produce delay effects of publication—treating stocks as “flows.” Next, define the state vector , where is max and is , with . Moreover, let be and be
where is quadrant of , and is block-row of conformable with the partition . Then the state equation can be written as:
with for and for . The observation equation is constructed in two steps—Step 1: Observation or measurement equation errors are partitioned conformable to , yielding . In the absence of cross-sectional aggregation, error-corrupted stocks, and flows, then
with , p.d., , for , . Step 2: is vector of values of for period , which are actually observed .
and , . Zadrozny (1990) studies changes in employment—the high-frequency (monthly) series that is treated as a stock variable whereas the second low frequency GDP is treated as a flow. More recent examples of mixed frequency VAR models include Kuzin, Marcellino, and Schumacher (2011), Schorfheide and Song (2015), and Eraker, Chiu, Foerster, Kim, and Seoane (2015), among others.
A second example was introduced by Mariano and Murasawa (2003). It contains a monthly AR factor, a static factor representation for monthly indicators, and an unobserved monthly GDP growth. They consider four monthly series: EMP, IP, Personal Income minus transfers (INC), Manufacturing and Trade Sales (SLS). In addition, the model contains a very important time-aggregation constraint, which helps to relate the observed quarter-on-quarter GDP growth to unobserved month-on-month GDP rate of change. The model is therefore as follows:
where is scalar latent factor with AR(p) structure:
monthly lag operator, and idiosyncratic AR(q) shocks:
The process is (hidden) monthly GDP whereas the process is a monthly observable process. Uppercase variables are level series, and lowercase ones are log growth rates. Superscript stars refer to latent processes. Then:
Moreover, is of dimension (typically equal to one) and is of dimension with . The state space representation is then as follows:
Assuming , and we need four lags of because is linear function of through yields:
We can then write the measurement equation as:
The Aruoba, Diebold, and Scotti (2009) model, yielding the so called ADS index, expands on the aforementioned model proposed by Mariano and Murasawa (2003). They work with a dynamic factor model, treating business conditions as an unobserved variable, related to observed indicators. Aruoba, Diebold, and Scotti (ADS) explicitly incorporate business conditions indicators measured at different frequencies. Important business conditions indicators do in fact arrive at a variety of frequencies, including quarterly (e.g., GDP), monthly (e.g., industrial production), weekly (e.g., employment), and continuously (e.g., asset prices). In particular, they explicitly incorporate indicators measured at high frequencies, given that their goal is to track the high-frequency evolution of real activity. Finally, ADS extract and forecast latent business conditions using linear yet statistically optimal procedures, which does not involve approximations.
So called bridge equations are linear regressions that link (“bridge”) high-frequency variables, such as industrial production or retail sales, to low-frequency ones, for example, the quarterly real GDP growth, providing some estimates of current and short-term developments in advance of the release. The “Bridge model” technique allows computing early estimates of the low-frequency variables by using high-frequency indicators. They are not standard macroeconometric models, because the inclusion of specific indicators is not mainly based on causal relations, but more on the statistical fact that they contain timely updated information.
In principle, bridge equation models require that the whole set of regressors should be known over the projection period, but this is rarely the case. Taking forecasting GDP as an example, because the monthly indicators are usually only partially available over the projection period, the predictions of quarterly GDP growth are obtained in two steps. First, monthly indicators are predicted over the remainder of the quarter, usually on the basis of univariate time series models (in some cases VAR have been implemented in order to obtain better forecasts of the monthly indicators), and then aggregated to obtain their quarterly correspondent values. Second, the aggregated values are used as regressors in the bridge equation, which allows the obtainment of forecasts of GDP growth.
It will be convenient to focus on a mixture of two frequencies, respectively high and low. In terms of notation, indexes the low-frequency time unit, and is the number of times the higher sampling frequency appears in the same basic time unit (assumed fixed for simplicity). For example, for quarterly GDP growth and monthly indicators as explanatory variables, . The low-frequency variable will be denoted by , whereas a generic high-frequency series will be denoted by where is the (past) high-frequency period with . For a quarter/month mixture one has , , as the last, second to last, and first months of quarter Obviously, through some aggregation scheme, such as flow or stock sampling, we can always construct a low-frequency series We will simply assume that .1
Let us start with a static regression model:(3.1)
where is an error term assumed to be identically independently distributed (i.i.d.) and the parameters can be estimated via ordinary least squares (OLS), yielding and Suppose now we want to predict the first out-of-sample (low-frequency) observation, namely:
Unfortunately, we do not have , but recall that we do have high-frequency observations available. For example, if we have values for for the first two months of quarter ( and ), then only is missing to complete the quarterly value Using an empirically suitable univariate time series model applied to the high-frequency observations, we can obtain:
where is a polynomial lag operator for one high-frequency horizon prediction with parameter estimates obtained over a sample of size (as we also have two months for quarter ) and .
We can index with the forecast horizon in high frequency, to emphasize that it does depend on the forecast horizon (the index was dropped when in the previous formula). Then, in general for consecutive high-frequency observations we can replace the unknown regressor with partial realizations of the high-frequency observations and complemented with estimates of the missing ones, namely the missing ones with high-frequency based predictions, namely:(3.2)
for . These are a collection of so called nowcasts (or flash estimates in statistical terminology), using bridge equations.2
The discussion so far brings us to mixed-data sampling (MIDAS) regressions. Unlike bridge equations, MIDAS regressions do not require a two-step procedure that consists of estimating a low-frequency and high-frequency data model separately. Schumacher (2016) provides a detailed discussion of the connection between bridge and MIDAS regressions.
MIDAS regressions are essentially tightly parameterized, reduced form regressions that involve processes sampled at different frequencies. The response to the higher-frequency explanatory variable is modeled using highly parsimonious distributed lag polynomials, to prevent the proliferation of parameters that might otherwise result, as well as the issues related to lag-order selection.
The basic single high-frequency regressor MIDAS model for -step-ahead (low-frequency) forecasting, with high-frequency data available up to is given by:(3.3)
where , and .
The parameterization of the lagged coefficients of in a parsimonious way is one of the key MIDAS features. Various other parsimonious polynomial specifications have been considered, including (1) beta polynomial, (2) Almon lag polynomial specifications, and (3) step functions, among others. Ghysels, Sinko, and Valkanov (2006) provide a detailed discussion.3 One of the most used parameterizations is the one known as “Exponential Almon Lag,” because it is closely related to the smooth polynomial Almon lag functions that are used to reduce multicollinearity in the distributed lag literature. It is often expressed as(3.4)
This function is known to be quite flexible and can take various shapes with only a few parameters. These include decreasing, increasing, or hump-shaped patterns. Ghysels, Santa-Clara, and Valkanov (2006) use the functional form with two parameters, which allows a great flexibility and determines how many lags are included in the regression.
Another possible parameterization, also with only two parameters, is the so-called Beta Lag, because it is based on the Beta function:(3.5)
and . One attractive specific case of the MIDAS Beta polynomial involves only one parameter, namely setting and estimating the single parameter with the restriction that it be larger than one, which yields single-parameter downward sloping weights more flexible than exponential or geometric decay patterns.
These are among the most popular parameterizations besides U-MIDAS and MIDAS with step functions. The parameterizations described are all quite flexible. For different values of the parameters, they can take various shapes: weights attached to the different lags can decline slowly or fast, or even have a hump shape. Therefore, estimating the parameters from the data automatically determines the shape of the weights and, accordingly, the number of lags to be included in the regression.
Suppose now, we want to predict the first out-of-sample (low-frequency) observation, namely considering equation (3.3) with :(3.6)
where the MIDAS regression model can be estimated using nonlinear least squares (NLS); see Ghysels, Santa-Clara, and Valkanov (2004) and Andreou, Ghysels, and Kourtellos (2010) for more details. Nowcasting, or MIDAS with leads as coined by Andreou, Ghysels, and Kourtellos (2013), involving equation (3.6) can also be obtained. For example, with additional observations the horizon shrinks to , and (3.6) becomes:
where we note that all the parameters are horizon specific. Therefore, the MIDAS regression needs to be re-estimated specifically for each forecast horizon. In other words, for a given choice of , we will obtain different estimates of the model parameters, because we are projecting on a different information set (as usual in direct forecasting). Therefore MIDAS regressions always yield direct forecasts.
Because autoregressive models often provide competitive forecasts compared to those obtained with static models that include explanatory variables, the introduction of an autoregressive term in the MIDAS model is a desirable extension.
Andreou, Ghysels, and Kourtellos (2013) introduce the class of autoregressive distributed lag MIDAS regressions, or ADL-MIDAS regressions, extending the structure of autoregressive distributed (ARDL) models to a mixed frequency setting. Assuming an autoregressive augmentation of order one, the model can be written as:(3.7)
Hence, an ADL-MIDAS regression is a direct forecasting tool projecting a low-frequency series, at some horizon , namely onto (or more lags if we consider higher order AR augmentations) and high frequency data Nowcasting, or MIDAS with leads, can again be obtained via shifting forward the high-frequency data with increments. The parameters are horizon specific and the forecast is one that is direct (instead of iterated).
Foroni, Marcellino, and Schumacher (2015) study the performance of a variant of MIDAS that does not resort to functional distributed lag polynomials. In the paper, the authors discuss how unrestricted MIDAS (U-MIDAS) regressions can be derived in a general linear dynamic framework, and under which conditions the parameters of the underlying high-frequency model can be identified; see also Koenig, Dolmas, and Piger (2003).
Suppose is small, like equal to three—as in quarterly/monthly data mixtures. Instead of estimating in equation (3.7) let us estimate the individual lags separately—hence the term unrestricted—yielding the following MIDAS regression:(3.8)
which implies that in addition to the parameters and we estimate additional parameters. With and small, like, say, up to four (annual lags) and large enough to make the error term uncorrelated, then, all the parameters in the U-MIDAS model can be estimated by simple OLS. From a practical point of view, the lag order could differ across variables, and and could be selected by an information criterion such as Akaike Information criteria (AIC), Schwarz information criteria (SIC), or Hannan-Quinn criteria.
The U-MIDAS regression has all parameters unconstrained and therefore runs against the idea that high-frequency data parameter proliferation has to be avoided. That is why U-MIDAS works only for small values of Is there an intermediate solution where we keep the appeal of simple OLS estimation, avoiding the nonlinear estimation setting of typical MIDAS regressions, and still keep the number of parameters small? The solution to this is called MIDAS with step functions, as introduced by Ghysels, Sinko, and Valkanov (2006) and Forsberg and Ghysels (2006). A MIDAS regression with steps and lags can be written as:(3.9)
where . Hence, we only estimate parameters for the high-frequency data projection with . The indicator function applies parameters to segments of high-frequency data lags past and prior or equal to The appeal is obvious, as we approximate the smooth polynomial lags via discrete step functions. Model selection amounts to selecting the appropriate set of steps, which can be again guided via information criteria. A popular application of MIDAS with step functions is the so called HAR model of Corsi (2009) involving daily, weekly- and monthly realized volatility.
To allow for the inclusion of several additional explanatory variables into the MIDAS framework, it is necessary to extend the basic model (3.9) as follows:
where is the number of high-frequency series. Within the more general framework, it is also possible to include explanatory variables at different frequencies, given that each indicator is modeled with its own polynomial parameterization. As an example, quarterly GDP growth can be explained not only by monthly indicators but also by weekly financial variables, with the explanatory variables, therefore, sampled at two different frequencies. Generically, this includes regression models with different high frequencies, say :
Obviously, this specification may be extended to allow for the presence of an autoregressive structure.
In practice, adding explanatory variables substantially complicates estimation. An alternative, easier procedure, is to work with single indicator MIDAS models and then pool the resulting forecasts. This approach works well, for example, in an empirical application on nowcasting U.S. GDP growth; see Kuzin, Marcellino, and Schumacher (2013). Alternatively, Andreou, Ghysels, and Kourtellos (2013) use time varying forecast combination rules to handle large data of daily financial market time series.
Breitung and Roling (2015) introduce a nonparametric MIDAS regression, which they use to forecast inflation with daily data. The model can be written as:
where instead of imposing a polynomial specification, Breitung and Roling propose an alternative nonparametric approach that does not impose a particular functional form but merely assumes that the coefficient is a smooth function of j in the sense that the absolute values of the second differences are small.
Various MIDAS regression models involving asymmetries or other nonlinearities have been proposed in the context of volatility forecasting and are covered in Ghysels and Marcellino (2018). Also originating in the volatility literature, but of general interest, is the semi-parametric MIDAS regression model of Chen and Ghysels (2011):
where is a function estimated via kernel-based nonparametric methods. Hence, the time series dependence is a standard MIDAS polynomial and therefore parametric in combination with the estimation of a generic function. The asymptotic distribution of the estimation procedure has a parametric and nonparametric part. The latter is kernel-based, involves solving a so-called inverse problem, and is inspired by Linton and Mammen (2005). The mixed data sampling scheme in semi-parametric MIDAS regressions adds an extra term to the asymptotic variance compared to the result obtained by Linton and Mammen.
Galvão (2013) proposes a new regression model that combines a smooth transition regression with a mixed data sampling approach, which she calls STMIDAS. In particular, let us write equation (3.3) as follows:
where . Then the smooth transition MIDAS regression can be written as:
The transition function is a logistic function that depends on the weighted sum of the explanatory variable in the current quarter. The time-varying structure allows for changes in the predictive power of the indicators. When forecasting output growth with financial variables in real time, statistically significant improvements over a linear regression are more likely to arise from forecasting with STMIDAS than with MIDAS regressions, because changes in the predictive power of asset returns on economic activity may be related to business cycle regimes.
So far, we have seen models that take into account mixed-frequency data in a univariate approach. Now we focus on multivariate methods and in particular VAR models. Contributions to this literature include Anderson et al. (2015) and Ghysels (2016). In either case, both classical and Bayesian estimation has been considered. Ghysels (2016) introduces a different mixed frequency VAR representation, in which he constructs the mixed frequency VAR process as stacked skip-sampled processes. We will call the approach MIDAS-VAR to distinguish it from the parameter-driven approach. An example of an order one stacked VAR involving two series and with would be (ignoring intercept terms):(3.10)
Note that a bivariate system turns into a four-dimensional VAR due to the stacking. Moreover, the approach does not involve latent shocks/states or latent factors. This means there is no need for (Kalman) filtering. Technically speaking the approach adapts techniques typically used to study seasonal time series with periodic structures (see, e.g., Gladyshev, 1961). The innovation vector is obviously also of dimension . This means that each entry to the VAR has its own shock. Note that there are no latent high-frequency shocks to the low-frequency series. One implication is that we can apply standard VAR model techniques such as impulse response functions and variance decompositions.
When we examine the last equation in the system (3.10) we recognize a U-MIDAS regression model:
In contrast, the first equation (as well as the second and third) measures the impact of low-frequency on series, namely:
What about nowcasting? For this we need a structural VAR extension. Building on standard VAR analysis we can pre-multiply equation (3.10) with a lower triangular matrix and obtain the following system:
Reading from this system for the second equation, we have:
and for the last equation:
The latter is a MIDAS with leads equation we encountered in the discussions on nowcasting, whereas the former is a regression predicting high-frequency data in real time.
It should be clear by now that the MIDAS-VAR approach discussed here relies only on standard VAR techniques. Note that this also applies to estimation, which can be either classical or Bayesian, where the latter is appealing when the dimension of the VAR is large, which is easily the case with MIDAS-VAR systems due to the stacking of low- and high-frequency series.
McCracken, Owyang, and Sekhposyan (2015) assess point and density forecasts from an observation-driven MIDAS-VAR to obtain intra-quarter forecasts of output growth as new information becomes available, imposing restrictions on the MIDAS-VAR to account explicitly for the temporal ordering of the data releases. They show that the MIDAS-VAR, estimated via Bayesian shrinkage, performs well for GDP nowcasting: it outperforms the considered time series models and does comparably well relative to the Survey of Professional Forecasters. Bacchiocchi, Bastianin, Missale, and Rossi (2016) study how monetary policy, economic uncertainty, and economic policy uncertainty impact on the dynamics of gross capital inflows in the United States using a MIDAS-VAR. While no relation is found when using standard quarterly data, exploiting the variability present in the series within the quarter shows that the effect of a monetary policy shock is greater the longer the time lag between the month of the shock and the end of the quarter. In general, the effect of economic and policy uncertainty on U.S. capital inflows are negative and significant. Finally, the effect of the three shocks is different when distinguishing between financial and bank capital inflows from one side, and FDI from the other.
State Space Models and MIDAS Regressions
It is worth mentioning that Bai, Ghysels, and Wright (2013) establish a connection between state space models and mixed-data sampling (MIDAS) regressions. Or, put differently, between parameter-driven and observation-driven models. In particular, they consider a stylized state space model:
In addition at the end of each quarter , a second observation is available:
Using both low- and high-frequency past data yields a prediction formula:
where are steady state Kalman gains. This equation relates to the multiplicative MIDAS regression models. Note that the MIDAS regression setting is a reduced form that does not identify the individual parameters of the entire state space model equations. Nor does it recover the latent process, that is, it does not provide a straightforward solution to filtering. When combined with more sophisticated methods, like simulation-based indirect inference, it is possible to take advantage of MIDAS-type models to do filtering. See, for example, Gagliardini, Ghysels, and Rubin (2017).
The material presented here is in part based on recent surveys by Andreou, Ghysels, and Kourtellos (2011) and Foroni and Marcellino (2013). Bańbura, Giannone, and Reichlin (2011) and Bańbura, Giannone, Modugno, and Reichlin (2013) provide overviews with a stronger focus on Kalman filter-based factor modeling techniques.
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(2.) A flash estimate or nowcast is defined exactly as a preliminary estimate produced or published as soon as possible after the end of the reference period, using a more incomplete set of information than the set used for the final estimates.