Real-Time Forecasting with a Large, Mixed Frequency, Bayesian VAR

We assess point and density forecasts from a mixed-frequency vector autoregression (VAR) to obtain intra-quarter forecasts of output growth as new information becomes available. The econometric model is specified at the lowest sampling frequency; high frequency observations are treated as different economic series occurring at the low frequency. We impose restrictions on the VAR to account explicitly for the temporal ordering of the data releases. Because this type of data stacking results in a high-dimensional system, we rely on Bayesian shrinkage to mitigate parameter proliferation. The relative performance of the model is compared to forecasts from various time-series models and the Survey of Professional Forecaster's. We further illustrate the possible usefulness of our proposed VAR for causal analysis.


Introduction
Information available to forecasters changes over the forecasting horizon. This variation occurs because macroeconomic variables are observed at mixed frequencies (i.e., some variables are monthly, while others are quarterly) and are released at different times during the quarter, each with its specific publication lag. Thus, modeling the information flow in an efficient way is a central issue for practitioners and policymakers.
Many of the models that have been developed to deal with the variation in incoming information are estimated at the highest sampling frequency, treating the low frequency variables as latent processes. Typically these are modeled in a state-space framework and, consequently, are highly parameterized. In this paper instead, we address the forecaster's information problem by using a mixed-frequency Bayesian VAR estimated at the lowest common data frequency. More specifically, our VAR stacks the three quarterly observations of a monthly variable, modeling each individual monthly release with a separate process. In addition, we incorporate the data release properties with recursive restrictions imposed on the VAR. Ordering the data consistent with the release calendar helps us deal with the publication lags: We model the available information set and consider its predictive content with no particular emphasis to the reference period of the released data.
Our VAR is a high-dimensional variant of the model proposed in Ghysels (2015), evaluated in real-time for out-of-sample nowcasting and forecasting of U.S. real GDP growth. Given that our model is large and unrestricted relative to the various specifications considered in Ghysels (2015), we estimate it relying on Bayesian shrinkage. 1 Our implementation introduces some technical intricacies which we further discuss in the paper. As we also aim to postulate a model that can be used for policy analysis, we evaluate it according to two criteria: (i) its ability to provide reliable forecasts and (ii) its usefulness for causal analysis. We compare the predictions of our model to commonly used time series alternatives considered in the literature, as well as competitive consensus forecasts provided by the Survey of Professional Forecasters (SPF). In addition, we quantify the differential effects of discretionary monetary policy shocks on real economic activity depending on when they occur during the quarter.
There are a variety of alternative forecasting models that address the variation in data sampling and data release timings-e.g., state-space models and Mixed Data Sampling (MIDAS). State space models are usually specified at a high-frequency and filtering/smoothing algorithms provide updates for the missing values of low frequency variables. 2 MIDAS regressions (Ghysels et al., 2004) 1 See Foroni, Guérin and Marcellino (2013) for a classical evaluation of a small-scale stacked VAR for the Euroarea in real-time. Blasques et al. (2015) provides a weighted maximum likelihood estimation procedure for mixed frequency dynamic factor models considered in the stacked form. 2 Giannone et al. (2008), Aruoba et al. (2009), and others utilize state-space framework to estimate factor models. They summarize large panels of macro data with a few factors to obtain high-frequency updates for missing observations of (latent) variables of interest. Schorfheide and Song (2015) use a mixed-frequency medium-scale VAR to obtain monthly nowcasts and forecasts of quarterly GDP growth.
1 are typically implemented for univariate autoregressive distributed lag models, allowing the high frequency data to enter the predictive regressions by sparsely parameterized weighting polynomials.
Under certain non-restrictive conditions, VARs can be mapped back to these alternatives. 3 In fact, our forecasting model parallels with a multivariate version of MIDAS regressions where we use unrestricted, data-driven weights instead of weighting polynomials. Thus, which model performs the best can not be known ex ante and is an empirical question. 4 However, as discussed in Ghysels (2015), in the context of mixed frequency data there is a difference between state-space models and the proposed mixed-frequency, stacked VAR: Statespace models are parameter-driven whereas the stacked VAR is observation-driven. Furthermore, high-frequency VAR models with latent shocks yield infinite-order VAR models in the observables, while the stacked VAR is of finite order by construction. This misspecification could be relevant for causal analysis. Consequently, while the various mixed-frequency alternatives are expected to be similar in their forecasting performance, the stacked VAR may be more generally useful.
We find that the mixed-frequency nature of the data is important for nowcasting, but that not all intra-quarter information improves forecast accuracy. Our model dominates some competitive alternatives-including a variant of a factor model-when nowcasting real GDP growth. In fact, the accuracy of our model's nowcasts are comparable to those of the SPF's: The root-mean-squared forecast error associated with our nowcasts is only 7 percent higher than that associated with the nowcasts of the SPF. Admittedly, we find considerable time variation in the forecasting performance of the models. For example, our model performs better than the factor model during the post-Great Recession period, but the nowcasts of the SPF have been more accurate since 2000. Controlling for a few outliers by using a median-squared forecast errors loss criterion makes our model even more competitive relative to the SPF.
Finally, we show that our model can be useful for causal analysis in ways that a low frequency VAR cannot. Specifically, we quantify the effects of conventionally identified monetary policy shock on real output growth but do so allowing for the shock to occur at different months within the current quarter. This is potentially relevant insofar as historically, Federal Open Market Committee (FOMC) meetings have met in all three months of each quarter. We find that the monetary policy shocks with the most impact on real output growth occur in the second month of the quarter. These results could be interpreted in two ways: Shocks that occur earlier in the quarter have more time to propagate through the real economy or the marginal predictive content of the high-frequency variables are largest in the second month of the quarter.
The remainder of the paper is organized as follows. Section 2 lays out the econometric specification of the blocking model and discusses the data. Section 3 considers the estimation methodology. Section 4 presents the construction of nowcasts and forecasts; Section 5 discusses the forecasting results and compares them to various alternatives. Section 6 investigates the usefulness of our model for causal analysis. We conclude with Section 7.

The Setup
In this section, we describe the data and method for performing our out-of-sample forecasting exercises. In our description of the data, we highlight the timing of their releases. These release dates will be used to generate a number of restrictions on the contemporaneous effects in the VAR outlined in the following subsection. Table 1 summarizes the data. The choice of the data series is based on some judgment as to which economic variables are important and is broadly followed by markets and policymakers. 5 For each series listed in the first column, we indicate the original frequency of the data, as well as the transformation used to achieve stationarity. 6 We specify the approximate release date of the series, their publication lags, the dates of the first available vintages, as well as the economic data release report from which the series are obtained. The data are combined from ALFRED (ArchivaL Federal Reserve Economic Data) database of the St. Louis Fed and Haver Analytics (marked by a †). We have monthly real-time vintages from 1980:01 to 2014:04. However, in the estimation we use vintages starting from 1985:01 onwards to ensure we have sufficient degrees of freedom for our estimation. 7 The earliest observation for any vintage is 1971:01. 8

INSERT TABLE 1 HERE
To make the information reported in the table more transparent, take the example of employment labeled as "Empl.: Total Nonfarm Payrolls". "M" in the frequency column indicates the monthly nature of the employment data. Employment, with some exceptions, is released on the first Friday 5 For some discussion on data choice, see McBride (2014). 6 The transformations are broadly consistent with (unreported) unit root test results. The growth rates are annualized, consistent with quarterly frequency, while variables in percentage differentials are multiplied by 4.
7 While degrees of freedom is less important for Bayesian estimation, we start the sample in 1985:01 to minimize the impact of the prior on the estimation results. Panel B in Figure 1 shows that considerable amount of uncertainty remains even when starting the estimation in 1985:01. 8 Although the dataset is "real time" in the sense that the most recent vintage is used for estimation in each out-of-sample experiment, we do not forecast data-revisions. To accommodate data revisions, additional equations would have to be included to link the revisions across releases (especially for GDP). We leave this exercise for future research.

3
of each month and pertains to the previous month. Thus, employment series has one-month publication delay, which captures the difference between the release and reference months.
Over time, new real-time data series become available. As we attain real-time vintages for these variables, we incorporate them in our estimation. As a result, the dimensionality of our model changes six times across our collection of vintages. 9 Prior to 1996, our model is comprised of sixteen monthly and one quarterly variables. In the late 1990s, we add core CPI (1996:12), ISM Manufacturing PMI (1997:03), and New Single Family Houses Sold (1999:07). The latter two series are considered to be leading indicators for real activity. The core CPI, on the other hand, can improve the interest rate-and consequently real output growth-forecasts as it is central to vintage. We include these variables because of their timely release.
Since our objective is to model the timing of the releases explicitly, we collect the data into five groups based on the timing of their release dates using a few simplifying assumptions: 1. Group 1 consists of the variables that are released during the first week of the month; Group 2 consists of the variables released during the second and third weeks in the month; Group 3 consists of the real variables released during the last week of the month; Group 4 is the policy instrument; and Group 5 includes other financial variables. 10 2. The number of estimated parameters in our model increase faster than in the standard VAR of equal lag order. Introducing variables at higher than monthly frequencies (weekly or daily) will exacerbate this problem. In order to mitigate parameter proliferation, we consider data only in monthly and quarterly frequencies. Financial variables are summarized at a monthly frequency, using end of month values. The exception is the S&P 500 Stock Index, for which we use the average value of the month.
3. Initial Unemployment Insurance Claims is a weekly series and are usually revised substantially. Each weekly release reports the four-week-moving-average of initial claims. We use 9 Retail Sales is a splice of two series RETAIL (Retail Sales) and RSAFS (Retail and Food Services Sales). Post 2006:06, we use the RSAFS data and its revisions, always splicing the data with the historic values of RETAIL for pre-1992:01. Moreover, for Retail Sales, Housing Starts and New Houses Sold, there are vintages that provide revisions for historical data, but no new observations. In these cases, we proxy the new values by repeating the values of the previous month so that the last few values (depending on the number of missing observations) in the faux-vintage are identical. This is consistent with the assumption that we are modeling what is in the forecasters information set. 10 The release dates required some approximations. For example, the Employment Situation Report can be released on the second Friday of the month when the first Friday falls on the 1st of the month and when the prior month has fewer than 31 days. This happens rarely. In such cases, we treat the reports as in the table as the changes usually still preserve the temporal ordering. the first weekly release in the month which reflects the past month's average of initial claims.
Based on this assumption, Initial Unemployment Insurance Claims is placed in Group 1 with the variables that are released at the beginning of the month.
We assume that variables released in the same group are considered simultaneous in the Wold causal chain sense. This assumption may be reasonable from a policymaker's point of view; in many circumstances, the costs associated with acquiring and explaining daily forecasts far exceed the benefits. We also further assume that each group reacts to the groups released before but are independent of the groups released after.
We separated the federal funds rate from the rest of the financial variables to analyze the effects of monetary policy shocks throughout the quarter. By isolating the federal funds rate, we can combine causal restrictions of the conventional empirical monetary policy literature (as in Christiano et al., 1999) with temporal restrictions motivated by data releases. Specifically, we assume that the monthly federal funds rate responds to all the information from the previous month and to the information in non-financial variables in the current month. Thus, monetary shocks can affect the current quarter's real GDP growth, but monetary shocks occurring in the first month of the quarter cannot affect the real GDP growth reported in the same month. Real GDP growth released in the first month of the quarter refers to previous quarter's growth and is already realized before the monetary shock occurs. We discuss this further in Section 6.
It should be noted that our data are organized based on the release and not reference calendar.
Thus, in January, the information set that is used for forecasting typically includes data pertaining to the past month and the current month, as well as the past quarter's realization of GDP. We use all these information for predicting the release of the current quarter real GDP growth. Within a quarter, the exercise parallels the direct forecasting approach (which is more robust in the case of misspecification), rather than filling the missing observations consistent with some parametric model such as the state-space model.

Model
Our forecasting model is a VAR, estimated at the lowest sampling frequency. While the data releases are observed in mixed frequencies, we treat multiple releases in the lowest frequency as separate observations modeled in a blocked linear form consistent with the previously defined groups.
Without loss of generality, we define our notation consistent with the specification used in our empirical application, mixing monthly and quarterly variables; alternative frequency mixing is a straightforward extension. Let x t−τ represent the high frequency (monthly) variable and y t denote the low frequency (quarterly) variable, for quarters t = 1, ..., T and months within the quarter τ = {2/3, 1/3, 0}. In this setup, each τ represents an intra-quarter month: τ = 2/3 indexes data 5 that are observed during the first month of the t quarter and τ = 0 indexes data that are observed in the last month of the t quarter.
For illustrative purposes, consider the case with one quarterly variable (say, real GDP) and one monthly variable (say, payroll employment) that is released during each of the three intra-quarter months prior to the release of the quarterly variable. Define the vector Y t = [x t−2/3 , x t−1/3 , x t , y t ] ′ ; then, the mixed-frequency VAR takes the form: where A ′ and B ′ l are n × n parameter matrices, C is a vector of deterministic components, p is the lag length, ε t ∼ N (0, I), and T is the sample size in quarters.
The model structure is different from the standard VAR. Our model allows each monthly release of a data series within a quarter to have its own data generating process. If our data was only at a monthly frequency, the standard monthly VAR could be written in the stacked form with a set of restrictions on the coefficients across months of the quarter.
The matrix A ′ is used to impose temporal ordering restrictions on the elements of Y t depending on the timing of their releases over the quarter. In the example, the three monthly releases of x t−τ are observed in a specific order followed by the eventual release of the quarterly value of y t ; thus, n = 4 and A ′ is lower triangular (4 × 4) matrix. In this example, each release is assumed to occur on a completely different day and it seems reasonable to estimate a lower triangular A ′ .
If, on the other hand, the third monthly release of the high-frequency variable x t occurs on the same day as the quarterly release of y t , one can make various assumptions about the temporal relationship of these variables. We make the choice of leaving the contemporaneous impact of x t on y t unrestricted. 11 Accordingly, the A ′ matrix would have the following form: where a i,j -s, i, j = 1, ..., 4, are unrestricted scalars.
In general, we can write A ′ in terms of monthly blocks A ′ j , where j reflects the month of the quarter, and a denotes unrestricted matrices of varying dimensions: 12 11 Our forecasting problem is formulated such that yt is not updated in response to the xt because we assume they are released at the same time. 12 We omit the subindices and use a generic notation for simplicity.
Each monthly block, as suggested above, has its own internal timing. Based on the data and release date assumptions in Table 1, the (monthly) Wold causal chain relating the release information results in a monthly block lower triangular A ′ m , for m = 1, 2, 3 : Consequently, the dimension of A ′ m increases by one.

Estimation
In general, the dimension of VARs becomes large quickly; in our case, the parameter proliferation is exacerbated because adding one more monthly predictor adds three variables to the VAR. In order to address the dimensionality issue, we estimate the VAR with Bayesian methods assuming a shrinkage prior. Bańbura et al. (2010) show that Bayesian VARs forecast with sufficient accuracy, even when the considered systems are fairly large (e.g., 100+ variables).

The Prior
where n is the number of endogenous variables in the VAR, p is the lag order of the VAR and k = np + 1 is the number of parameters in each VAR equation. A general form of the VAR in equation (1) can be re-written as Let a i and f i represent the respective ith columns of A and F , accounting for the ith structural equation in the VAR. We consider a modification of the Sims and Zha (1998) prior. The prior imposes independence across the structural equations and assumes that the vectors a i and f i are both jointly normally distributed. More specifically, the prior postulates the following distributions on the parameters. For i = 1, ..., n, Sims and Zha (1998) consider a random walk prior which centers the reduced-form parameters on the first autoregressive lag at unity. Since we have transformed the variables to stationary, we instead center the reduced-form autoregressive lag coefficients on zero. Accordingly,P is parameterized as a matrix of zeros. Furthermore,S is a n × n diagonal positive definite matrix, where the elements on the ith row areS ij = .., n, and zero otherwise. On the other .., n, and zero otherwise. The prior variance for the constant, i.e., the kth diagonal is λ 2 0 λ 2 4 and uncorrelated with the rest of the parameters. For example, suppose n = 2 and p = 2. It follows from the discussion above that A scaling factor, σ 2 j , is included to mitigate the effect of a difference in the unit of measure across the variables. In practice, σ 2 j is proxied by the sample variance of the residuals that result from a univariate autoregression of order p for series j. The role of the hyperparameters λ 0 , λ 1 , λ 3 , λ 4 and λ 5 is presented in Table 2. 13

INSERT TABLE 2 HERE
Hyperparameters can be selected by borrowing the values from, for example, Sims and Zha (1998) or Clark and Doh (2014). However, we mix the monthly and quarterly series together differently from previous studies. Moreover, as shown in Bańbura et al. (2010) the optimal level of shrinkage increases as the dimensions of the model increase relative to the sample size. Therefore, it is desirable to allow hyperparameter values to be selected in a systematic way.
We compute the marginal data density to choose the hyperparameter values λ 0 and λ 1 (hence, their values are "tbd" in Table 2). We set λ 3 = 1, the same value used by Bańbura et al. (2010), and set λ 4 = 1, consistent with Sims and Zha (1998). In principle, we could also use the marginal data density to obtain the values for λ 3 and λ 4 , but, given that we are conducting a recursive forecasting exercise, this would increase computation time. Moreover, according to Giannone et al. To make the out-of-sample forecasting exercise easy to conduct, an analytical solution for the marginal data density is preferred. As our causal and temporal restrictions result in an underidentified system, a closed-form solution for the marginal data density exists and is presented below: 15 where

The Sampler
The model is estimated via Gibbs sampler proposed in Waggoner and Zha (2003). 17 We need to estimate the VAR in equation (5) subject to constraints imposed by matrices Q i and R i . The restrictions are imposed such that for i = 1, ...., n: 14 We estimated the model by choosing all hyperparameter values from the marginal data density. The results are not sensitive to those choices. 15 Details are provided in Appendix A and a numerically more stable implementation is discussed in Appendix B. 16 We tried using finer grids and larger intervals, yet it did not make too much of a difference for the results. In principle, we could also allow for different shrinkage for monthly and quarterly variables. In this case, the marginal data density calculations are not trivial. Whether or not the adjustment of the shrinkage would make a difference for forecasting is an empirical open question that we leave for future work. 17 There are alternative ways to estimate a VAR as the one in equation (5). We prefer Waggoner and Zha (2003) approach as it provides with an efficient algorithm to estimate a VAR even when the contemporaneous correlation matrix A is overidentified, resulting in a restricted reduced form variance-covariance matrix for the VAR. Since we have modeled the within group releases of data as contemporaneously correlated, A is underidentified in our case. Nevertheless, we prefer to utilize a general framework that is not restrictive in terms of the form of temporal restrictions one can impose. If one imposes many restriction such that the system becomes overparameterized, then the marginal data density calculations in the Appendices A and B become approximations.
For our implementation Q i is a n×n matrix parameterized to incorporate the temporal ordering restrictions highlighted in Section 2, while R i is a k × k matrix of zeros reflecting the fact that there are no restrictions on the lag dynamics. We further define U i and V i to be matrices whose columns form an orthonormal basis for the null spaces of Q i and R i . We can then rewrite our parameter space in terms of ψ i = U i f i and φ i = V i g i , where the orthonormal rotation matrices U i and V i reduce the parameter space of the VAR, taking into account the linear restrictions on the contemporaneous and lagged dynamics (which is unrestricted in our case) of the system. (2003) show, the prior in equation (6), the likelihood of the model in equation (5) and the restrictions in equation (8) imply marginal posterior distributions for ψ i and φ i represented by:

As Wagonner and Zha
where H, P , and S are transformations of the prior mean and variance matricesH,P , andS and ϕ(.) denotes the pdf of a normal distribution.
The implied joint conditional posterior distribution of ψ i s is independent of φ i s, but nonstandard. Waggoner and Zha (2003) propose a Gibbs sampler as a strategy to simulate from this joint distribution. Starting from some arbitrary initial values of {ψ * 1 , ..., ψ * i−1 , ψ * i , ψ * i+1 , ..., ψ * m }, Gibbs sampler sequentially draws from the conditional posteriors p(ψ i |ψ 1 , ..., ψ i−1 , ψ i+1 , ..., ψ m , X, Y) as a mechanism to draw from the joint. In order not to be redundant and to conserve some space, we refer the reader to Waggoner and Zha (2003) for the details on how to draw from the conditional distributions. Given the simulated values of ψ i s, simulating the φ i s is fairly easy as they are drawn from a normal distribution.
As Waggoner and Zha (2003) suggest, the algorithm results in Monte Carlo chains that mix fairly well and converge fairly fast. Thus, our reported results are based on 5,000 accumulated. 18 Once we have the distribution of φ i s and ψ i s, the distributions of a i s and f i s, thus of A and F , could be easily recovered with the help of the orthonormal rotation matrices U i and V i .

Forecasting
To document the forecasting performance of our proposed model in real-time, we consider point and density forecasts, focusing on nowcast and four-quarter-ahead (one-year-ahead) forecasts. To simplify the discussion that follows, we first address some timing issues: 1. January, April, July, and October are the months for which we have observations for the previous quarter real GDP growth and all other variables in our dataset.
2. The model is only estimated in January, April, July, and October after a new GDP release is obtained. The posteriors of the parameter estimates are stored to construct intra-quarter forecasts. The lag length is fixed at p = 4 for the entirety of the empirical implementation.
3. We compute two forecasts, Y t+1 and Y t+4 . During the first month of the quarter, the onequarter-ahead forecast is effectively forecasting the advance release for first quarter real GDP growth. Similarly, Y t+4 is a one-year-ahead forecast.

Point Forecasts
There is always a question of which measure of central tendency to consider when dealing with optimal forecasts. Gneiting (2012) argues that when the scoring function is in a form of a squared error, the optimal point forecast is the mean of the predictive distribution. Therefore, we construct point forecasts for all 5000 draws of A and F and report the mean of the forecast distributions as a point forecast. This is equivalent to reporting the conditional mean of Y t+h , h = 1, 4, for each parameter draw given the information available at time t and immediately following each group of releases.

11
We provide the exact formulas for forecast construction after every monthly release; they can be easily adjusted for more granular forecasts. Let n 1 , n 2 and n 3 , be the number of variables released in Month 1, Month 2 and Month 3, respectively, where n 1 + n 2 + n 3 = n. For each parameter draw, nowcasts are obtained as: 1. Time t (Month 1): At time t, E(ε t+h |X t , Y t ) = 0 for all h > 0 and hence: 2. Time t + 1/3 (Month 2): Let i 1/3 = [1 n 2 ×n 2 , 0 n 2 ×n 3 , 0 n 2 ×n 1 ] ′ .
With a logic similar to that in the previous step we get:

Predictive Distribution
In addition to point forecasts, we are also interested in the evaluation of density forecasts. The predictive density for h = 1 is defined as: where p(F |A, X t , Y t ) and p(A|X t , Y t ) are the posterior distributions of the parameters F and A.
In order to construct the predictive distributions, we follow the algorithm below: 1. Draw 500 observations of ǫ t .
2. Given the posteriors of p(F |A, X t , Y t ), p(A|X t , Y t ), we construct the correspondingŶ t+1 for the forecasting model in equation (5).
3. In total, we have 5000 parameter draws. Thus, for each observation of ǫ t , we construct averages of the predictionŶ t+1|t across the draws.
4. Calculate percentiles for the distributions with 500 observations. 5. When calculating the predictive densities within the quarter, we simulate only the shocks that have not yet been realized. In other words, when computing density forecast at the end of month two, we simulate only month three shocks.

Evaluation
We use a few different evaluation criteria for our forecasting exercise. 19 First, we evaluate the unconditional predictive performance of our model (and the alternatives) using the root-meansquared forecast error for an out-of-sample evaluation period P = T − R − h: Next, we compute the root-median-squared forecast error to obtain a performance measure.
To compare our model to alternatives that are at time-smoothed consensus forecasts, we want an measure that is robust to outliers. The root-median-squared forecast error is: In addition, we consider conditional predictive ability tests by tracing measures in equations (15) and (16) over time with a fixed, rolling window of 10 years. To determine how the predictive densities change over time, we examine the evolution of the distance between the 97.5th and 2.5th percentiles over time.
We compare he forecasting performance of our model to some alternative time-series models and the Survey of Professional Forecasters' forecasts. For ease of exposition, we concentrate on the forecasting performance of four macroeconomic variables: employment growth ("EMP"), CPI inflation ("CPI"), the change in the Federal Funds Rate ("FFR") and real GDP growth ("GDP"); our emphasis throughout will be on the growth rate of real GDP.

INSERT FIGURE 1 HERE
First, Figure 1  Moreover, it appears that within-quarter information makes the distribution more concentrated around the mean.

Quarterly Models
We first evaluate the predictive performance of our quarterly model, ignoring the within-quarter monthly releases, and compare the forecasting performance of our model for h = 1 in Month 1 of each quarter to univariate autoregressive models and a simple VAR estimated with OLS. The intent of this exercise is to compare the predictive performance of our large VAR to that of parsimonious time series models, known to be competitive benchmarks for forecasting exercises.
The autoregressive models ("AR-Quarterly") are an AR(4) for the real GDP growth and an AR(12) for the monthly variables. We estimate monthly and quarterly ARs with data from the January, April, July and October vintages. For the monthly variables, we construct three-stepahead forecasts (with no new information) and compare the quarterly average of the monthly forecast to the quarterly average of the monthly realization. We construct forecast errors based on the averages of three intra-quarter monthly forecasts and realizations to make them comparable with our Bayesian VAR ("BVAR-Quarterly"). For the BVAR, we obtain three direct, one-stepahead forecasts per quarter (as the monthly variable appears in the blocking model three times over the quarter) and average them to obtain a quarterly value. For the single-frequency quarterly VAR ("VAR-Quarterly"), the model is a four-variable VAR(4) in employment growth, CPI inflation, changes in the Federal Funds Rate and real output growth as a forecasting model. 20 We average the monthly variables to a quarterly frequency by considering simple averages across the months in a quarter.
In Table 3, we report the rMSE for the "AR-Quarterly" (in bold); for the other the models, we report the ratio of their respective rMSEs to that of the "AR-Quarterly". Ratios less than 1 (marked in italics) indicate improvements. As the table suggests, the performance of the small-scale VAR ("VAR-Quarterly") is inferior to the autoregressive model. For employment growth and CPI inflation, the performance of our "BVAR-Quarterly" is also inferior to that of the autoregressive benchmark. However, our quarterly BVAR does as good as the parsimonious AR(4) for the real GDP growth. This is an encouraging result, since our estimated system is fairly large. For the changes in the FFR, our "BVAR-Quarterly" provides a 15 percent improvement over the AR.  where we extract factors from a monthly dataset and use these monthly factors to forecast the quarterly real GDP growth. For t = 1, ..., T and τ = {2/3, 1/3, 0}, the model is constructed as follows:

Mixed Models
where X t−τ is a (n − 1)/3 × 1 vector of monthly variables, F t−τ is a r × 1 vector of common factors and Y t is the real GDP growth. The parameters Λ, Π, α and β capture the parameters, while ξ t−τ and u t−τ are Gaussian white noise error terms. We estimate the number of factorsF t−τ via principal components, choosing the number of factors r such that they jointly explains at least 50 percent of the variation contained in the (n − 1)/3 macroeconomic data series at any t. 21

INSERT FIGURE 2 HERE
We estimate the model at the end of every month. Our forecast of the quarterly growth rate of the real GDP is determined by its projection over monthly factors at the end of each quarter.
In order to get the value of the extracted factor at the end of each quarter, we consider equation (18). Thus, at the end of t − 1, we consider a three-step-ahead, iterated forecast of the factor to get to a time-t value. Accordingly, at times t − τ , τ = 2/3, 1/3, we consider two-and one-step-ahead forecasts, respectively. The difference between how we structure the factor model relative to the traditional approaches is that we still keep the data ordered based on the release calendar, not the reference calendar, to make the results from the various considered approaches comparable. The

Survey of Professional Forecasters
The Survey of Professional Forecasters (SPF) provides forecasts of real GDP growth as well as a variety of economic fundamentals at the quarterly frequency. These include nowcasts as well as forecasts up to four quarters ahead. We use the forecasts of (annualized) quarterly real GNP/GDP growth rates. The series start in 1968:IV. We use the mean forecast across 34 professional forecasters starting 1985:I provided by the Federal Reserve Bank of Philadelphia. The choice of the mean is motivated with our loss function of squared forecast error, as well as with the intention to keep our model forecasts (based on the means of predictive distributions) comparable with that of the survey. We should also note that SPF is conducted with the advance release of the GDP as a target variable in mind, putting it on fair grounds with the forecasting exercise conducted in our paper. 22

INSERT TABLE 6 HERE
The timing of the SPF is crucial since it would be important to match the timing of our forecasts with that of the SPF. Usually, the professional forecasters respond to the survey in between the 22 Certainly, there are other competitive benchmarks one can use. For instance, Greenbook/Teelbook forecasts are the staff forecasts of the Federal Reserve Board and Blue Chip Economic Indicators (BCEI) are monthly professional forecasts that are sometimes considered as benchmarks. The Greenbook/Teelbook forecasts are available to the public with a five-year lag, while the BCEI are proprietary forecasts. Thus, we settle for SPF available at https://www.philadelphiafed.org/research-and-data/real-time-center/survey-of-professional-forecasters/datafiles/rgdp. SPF is known to be a competitive benchmark as suggested by the results in Giannone et al. (2008).
second and third weeks of the month. 23 Consistent with the temporal restrictions discussed in Section 2, the forecasts obtained after either Group 1 or Group 2 data are released match with the SPF timing the best. Table 6 reports the rMSE's for nowcasts constructed multiple times throughout the quarter, after each groups' data (as in Table 1) has been released. Group 5 coincides with the end of the month. We also report the rMSE for the SPF's mean forecast for real output growth. As the table suggests, the nowcasting performance of our model improves as the first two groups of data in the second month of the quarter become available. After that, the marginal impact of each group on the nowcast is negligible. The rMSE of our BVAR is 1.44 after the Group 2 data in the second month is released. This is only about 7 percent worse than that of the SPF with the rMSE of 1.35. Figure 3 sheds further light on the importance of the group releases for the performance of our BVAR relative to the SPF. We divide the figure into two parts: Panel A shows the smoothed rMSE-s calculated with a fixed rolling windows of ten years for the nowcasts obtained at the end of Month 1 and after the release of two group variables. As a reminder, the first group captured mainly soft survey data and the information from the Employment Situation Report, while group two had more diverse information such as industrial production, inflation and housing information as well as Philadelphia Fed's Business Outlook Survey. We show the behavior of the nowcasts after marginal releases of the 10 sequential groups in Panel B. In order to make the picture legible, we only plot the bounds corresponding to the point-wise maximum and minimum across the 10 various nowcasts. First, Panel B confirms the results of Table 6: there is not much action after the Group 2 in the Month 2 has been realized. As we can see from the figure, the bounds are narrow and they tightly replicate the behavior of the Group 2 nowcast presented in Panel A. Panel A, on the other hand, shows a few pieces of interesting information. First, our nowcasts are in general worse than that of the SPF even when we consider the nowcasting performance conditionally over time and not on average. There is a brief period in time between 1996 and 2002 that our model performs similar to the SPF, but this corresponds to a period where the SPF is particularly bad relative to other periods. Figure 4 shows the nowcasts based on our BVAR relative to the SPF based on a rMedSE criterion. As suggested by the figure, if one uses a median, which is robust to outliers, rather than a mean, then the relative ranking of the nowcasts changes. In fact, Panel A shows that according to rMedSE criterion our nowcasting model has done better than the SPF for most of the out-of-sample, though the performance has declined recently since mid-2005. Panel B suggests more variability in the marginal impact of data releases for the data that becomes available after the mid-second month of the quarter. Figure 5 shows the boxplot of the nowcasts from the BVAR after the Group 2 data has been released. The figure shows the median for forecast errors grouped by a rolling window of ten years worth of observations. In addition, it shows the 25th and 75th percentiles. Points are marked as outliers (by red +) if they are larger than Q3+1.5*(Q3-Q1) or smaller than Q1-1.

Conclusion
We present a stacked VAR as an alternative for high dimensional forecasting. We show that our VAR, estimated by Bayesian shrinkage, performs well for GDP nowcasting: it outperforms the considered time series models and does comparably well relative to the Survey of Professional Fore-casters' forecasts. Nevertheless, there is considerable time variation in the forecasting performance.
We show the usefulness of out setup for causal analysis by looking at the real GDP responses to discretionary monetary policy shocks throughout the quarter.
where Y t is an n × 1 vector of endogenous variables at time t, X t is a k × 1 vector of predetermined and exogenous variables at time t, , and a i and f i are the respective ith columns of A and F .
We can write the reduced form as: and e ≡ vec(ǫA −1 ), where vec(.) vectorizes the respective matrices making (tn × 1) vectors. Let The system can be rewritten as: The likelihood function is: 26 Since the prior imposes independence across the structural equations, we can re-write the priors in equation (6)  With the property of the trace operator tr(ABC) = vec(A ′ ) ′ (I ⊗ B)vec(C), we can rewrite: Note that we derive the marginal data density for an arbitrary prior characterized byP as opposed to it being a matrix of zeros as it is in the case of our empirical application. In the case of zeros the expressions simplify.
The marginal likelihood or marginal data density is then given by We can plug in the expressions of the respective probabilities to get: The last integral could be simplified. We proceed by simplifying the expressions under the exponentials separately.
The first expression under the exponential could be written as: The second expression under the exponential could be written as: The marginal data density becomes: We can rewrite the last integral as: where λ is a 1 × kn vector and Σ is a kn × kn matrix.
The integral could be written in the following form: This integral is a Gaussian integral and is equal to (2π) kn 2 |Σ| −1/2 exp 1 2 λΣ −1 λ ′ Let us simplify some relevant expressions that enter this integral. First, 27 |Σ| = |I n ⊗ (X ′ X +H −1 )| = |I n | k |X ′ X +H −1 | n = |X ′ X +H −1 | n 27 If A is a n × n matrix and B is a m × m matrix. Then, |A ⊗ B| = |A| m |B| n .
Further, let Ξ = (X ′ Y +H −1′P ). We can rewrite λ as: 28 Thus, the Gaussian integral becomes:  Notes: We report the out-of-sample rMSE for the AR-Quarterly model (in bold), while for the VAR-Quarterly and BVAR-Quarterly we report the rMSEs relative to that of the AR. Numbers in italics indicate improvements. Notes: We report the out-of-sample rMSEs for the AR model (in bold), while for the mixed-BVAR we report the rMSE relative to that of the AR. "BVAR -M1," "BVAR -M2" and "BVAR -M3" correspond to forecasts obtained at the end of Month 1, Month 2 and Month 3 in the quarter. Notes: We report the out-of-sample rMSEs for the "BVAR -M1," "BVAR -M2"and "BVAR -M3". Notes: We report the out-of-sample rMSEs for the mixed frequency nowcasts of the BVAR after each group's data (as in Table 1) has been released. We also report the rMSE for the SPF's mean forecast.        Notes: We show the cumulative response of the FFR and the real GDP to 25 bases point increase in FFR in Month 1, 2 and 3 of the quarter. The shaded area is the 95% coverage area associated with Month 1 monetary policy shock.