CCapturing GDP Nowcast Uncertainty inReal Time
Paul Labonne ∗ Abstract
Nowcasting methods rely on timely series related to economic growth forproducing and updating estimates of GDP growth before publication of officialfigures. But the statistical uncertainty attached to these forecasts, which iscritical to their interpretation, is only improved marginally when new dataon related series become available. That is particularly problematic in timesof high economic uncertainty. As a solution this paper proposes to modelcommon factors in scale and shape parameters alongside the mixed-frequencydynamic factor model typically used for location parameters in nowcastingframeworks. Scale and shape parameters control the time-varying dispersionand asymmetry round point forecasts which are necessary to capture theincrease in variance and negative skewness found in times of recessions. Itis shown how cross-sectional dependencies in scale and shape parametersmay be modelled in mixed-frequency settings, with a particularly convenientapproximation for scale parameters in Gaussian models. The benefit of thismethodology is explored using vintages of U.S. economic growth data with afocus on the economic depression resulting from the coronavirus pandemic.The results show that modelling common factors in scale and shape parametersimproves nowcasting performance towards the end of the nowcasting windowin recessionary episodes.
Keywords: Nowcasting uncertainty, score driven models, dynamic common factors,volatility, skewnessJEL codes: C32, C53, E66
This paper proposes a novel means for deriving timely measures of GDP nowcastuncertainty using forecasting errors observed in series related to economic growth.Nowcasts can thus be communicated with an improved appreciation of the statisticaluncertainty attached to them. The approach relies on filtering and dynamic factortechniques for exploiting dependencies in a panel of time series released on differentfrequencies and asynchronously. While Gaussian dynamic factor models providea reliable and popular method for nowcasting, non-Gaussian techniques are usedincreasingly to model the distribution of conditional GDP growth in a more flexibleway. This paper sets out new techniques for updating nowcasting uncertainty inboth Gaussian and non-Gaussian frameworks. ∗ King’s College London and Economic Statistics Centre of Excellence, email:[email protected]. I thank Martin Weale for his supervision on this project. a r X i v : . [ ec on . E M ] D ec s the leading measure of economic growth GDP is a key variable affectingpolicy makers and economic agents’ decisions. But its comprehensiveness comes atthe cost of timeliness; it is published with a significant delay. Nowcasting methodsprovide a means for forecasting current-quarter GDP starting from the beginningof the quarter until its official release by using data related to economic growthbut released earlier and more frequently. By modelling the relationship betweenthese related series and GDP, the ‘targeted’ series, it is thus possible to update thecurrent-quarter GDP forecast, or nowcast, each time new observations on relatedseries become available.For an efficient use of nowcasts the uncertainty attached to them should beconveyed as well. But while nowcasting methods make use of data related toeconomic growth to improve point forecasts, the signal on forecasting uncertaintythese related series carry is exploited only partially. The model capturing cross-sectional dependencies in the data is typically applied only to conditional means(or locations), whereas forecasting uncertainty is derived from scale and shapeparameters. While modelling dependencies in conditional means may be enoughfor forecasting uncertainty to be propagated across series when related series havedynamic variance parameters, this channel is incomplete and takes place with apotential delay. The sharp downturn brought by the coronavirus pandemic and theincrease in forecasting uncertainty linked to it makes this issue particularly salient.To make full use of the information that related series carry on GDP nowcastinguncertainty this paper explores the benefit of using dynamic factor models for scaleand shape parameters, in addition to the one typically used for locations. Hencenew observations in the related series lead to timely updates in the point forecaststhrough the common factor in the locations and, importantly, to the uncertaintyattached to them as well through the common factors in the scales and shapes.Intuitively, if large prediction errors occur in a series related to GDP, and theseare associated with large prediction errors in GDP, then the dispersion attached tothe GDP nowcast should be adjusted accordingly. This is possible directly throughthe common factor in the scales. On the other hand, modelling dependencies in theshape parameters is useful to capture the asymmetry in prediction errors typicallyobserved at the onset of recessions : While there is an increase in the dispersion ofprediction errors, these are likely to be skewed towards negative values. The onsetof the Covid-19 pandemic is a good illustration of this fact. It was clear in April2020 that the first-quarter GDP growth in the US would be negative or very close tozero, but while the magnitude of the drop was uncertain, very large positive figureswere clearly improbable.The motivation for modelling non-Gaussian features, and a shape parameterin particular, is based on evidence that macroeconomic data in general are notnormally distributed (see for instance Haldane (2012) and Jensen et al. (2020)). Thefeatures of GDP growth’s distribution when conditioning on macroeconomic andfinancial variables are also likely to vary over the economic cycle. By conditioningon financial variables, Adrian et al. (2019) and Delle-Monache et al. (2020) notably2nd increased negative skewness accompanying a rise in volatility in the predictivedistribution of US GDP growth in times of recessions. In normal times, however,GDP growth is close to being conditionally normally distributed. These temporarydeviations from normality have important implications regarding the communicationof a model’s predictions. A forecaster should take into account the time-varyingshape of the uncertainty attached to a model’s forecast, with a particular attentionto downside risk.To model openly non-Gaussian features in the conditional distribution of economicdata this paper adopts the score driven methodology of Harvey (2013) and Creal et al.(2013). Score driven models provide a flexibility close to non-Gaussian state spacemodels while remaining easy to estimate and implement. They have been appliedsuccessfully to economic forecasting problems notably by Delle-Monache and Petrella(2017), Creal et al. (2014) and Gorgi et al. (2019). More recently, Delle-Monacheet al. (2020) use score driven methods to study macroeconomic downside risk byestimating time-varying location, scale and shape parameters. They use a largenumber of regressors related to financial conditions in a univariate setting. Theframework presented here can be seen as a mixed-frequency multivariate alternativeto their model.Some nowcasting models allow variance parameters to change over time, notablythrough stochastic equations or Markov-switching processes, but their dynamicsdo not exploit cross-sectional information. Models with stochastic volatility (as inAntolin-Diaz et al. (2017, 2020) for instance) can nevertheless give an improvedmeasure of nowcasting uncertainty through the common factor in conditional means.A large prediction error in a related series increases the expected volatility of thenext prediction error in this series. The dynamic in the common factor comes fromthese prediction errors; therefore, the larger their expected volatility, the largerthe expected volatility of the common factor. Since the common factor is themain component of GDP nowcasts, their dispersion increases as well. Modellingdependencies in the scale parameters provide a complementary and direct approachfor updating GDP nowcasting uncertainty.Separately, models with volatility (or scale) common factors are not new but sofar have not been used in a mixed-frequency setting (see Carriero et al. (2016, 2018,2020) and Huber (2016)). Gorgi et al. (2019) model a common scale component in ascore driven bivariate model, but they do so by constraining the scale to be identicalin both series and do not discuss the issue of temporal aggregation. The modelproposed here contributes to this literature on stochastic volatility by addressingopenly the temporal aggregation problem arising from modelling common volatilitycomponents in mixed-frequency and possibly non-Gaussian frameworks. In doing soit provides an approach specifically fitted for nowcasting which may be applied in awide range of dynamic factor models.Modelling common components in series aggregated and sampled at differentfrequencies raises temporal aggregation issues somewhat different for location, scaleand shape parameters. While Mariano and Murasawa (2003) popularised a precise3pproximation for conditional means which is applicable to location parameters,similar solutions for scale and shape parameters have not been discussed in theliterature.This paper explores two different strategies to tackle the temporal aggregationproblem of scale and shape parameters. First, it is shown how scale parameters maybe aggregated temporally in Gaussian models using a convenient approximation,thus providing an approach for modelling volatility common factors in a widerange of mixed-frequency dynamic factor models. The second approach consistsof aggregating monthly series into rolling-quarterly figures. The mismatch in thefrequency of aggregation is thus alleviated, and scale and shape common factors canbe modelled in a non-Gaussian setting.The mixed-frequency dynamic factor model presented in this paper exploits fourUS time series : GDP, industrial production, the index of working hours, and theWeekly Economic Index. The first three series are derived from real-time vintagesprovided by the Federal Reserve Bank of Philadelphia. The Weekly Economic Index(Lewis et al. (2020)) is a recent indicator released by the New York Fed in the aimof capturing and communicating the effect of the Covid-19 pandemic on the USeconomic activity in a timely way. GDP is quarterly while the other series aremonthly.Different specifications of the model are applied in a real-time setting to studythe effect of modelling common factors in scale and shape parameters on nowcastingperformance, with a particular interest in density forecasts and recessionary episodes.The results show that models with common factors in their scale and shape paramet-ers perform better than models with only location common factors towards the endof the nowcasting window, while providing competitive performances in other times.Scale and shape common factors prove to be particularly important to capturethe forecasting uncertainty generated by the coronavirus pandemic. Separately,modelling fat tails in series related to economic growth complicates the identificationof turning points in activity. A dynamic factor approach is used for exploiting dependencies across series inlocation, scale and shape parameters. A popular method for estimating dynamicfactor models consists of writing the model in state space form and using theKalman filter to evaluate the model’s log likelihood. The Kalman filter providesan efficient approach for estimating nowcasting models for two reasons. First, ithandles easily the missing values that arise from modelling jointly series aggregatedat different frequencies and released asynchronously. Secondly, predictions can bedecomposed into latent components, which notably can be used to capture secularchanges in addition to common factors (see Harvey (1989) and Durbin and Koopman(2012)). A comprehensive presentation of dynamic factor and state space methods4or nowcasting is given by Banbura et al. (2013).A limitation of the Kalman filter, however, is that it relies on the data beingconditionally normally distributed. Non-Gaussian features can be introduced usingimportance sampling methods but these can be computationally intensive. Alternat-ively, Creal et al. (2013) and Harvey (2013) derive a new class of filters relying onthe conditional score where the conditional distribution of the observations can arisefrom a wide range of families. An attractive feature of score driven models is thatthey provide a general framework for introducing time-variation and latent states inany parameter of the predictive distribution. But unlike state space models, scoredriven models are fully deterministic conditional on past information. Apart fromthis shortcoming they retain most of the flexibility of state space models, includingthe ability to model unobserved components. This section extends the score drivendynamic factor model of Creal et al. (2014) with dynamic factor structures for scaleand shape parameters. Hence new observations on related series lead to timelyupdates in both the point forecasts and the uncertainty attached to them.
As in Creal et al. (2014) and Gorgi et al. (2019) each element of the observationvector y t = ( y ,t , ..., y m,t ) (cid:48) , t = 1 , ..., N can have a distinct conditional (or predictive)density such that y i,t ∼ p i ( y i,t | Y t − ) , i = 1 , ..., m, (1)where m is the number of time series. The set Y t − = { f t , X t − , Θ } includes theinformation available at time t −
1. All variables are modelled at a monthly frequency.The GDP series, which is quarterly, shows a quarterly figure in the last month ofeach quarter with missing values in other months. The vector f t includes thedynamic states related to location, scale and shape parameters which are fullydeterministic conditional on past information. Θ is a set of fixed parameters suchas autoregressive coefficients and factor loadings. Importantly the observations areassumed to be cross-sectionally independent conditional on past information. Hencethe log likelihood of the observations at time t can be written aslog p ( y t | Y t − ) = m (cid:88) i =1 δ i,t log p i ( y i,t | Y t − ) , (2) t = 1 , ..., N , where δ i,t is zero if observation y i,t is missing and one otherwise. Following Creal et al. (2013) and Harvey (2013) the dynamic in the vector oftime-varying parameters f t comes from the conditional score such that f t +1 = Bf t + As t , (3)5here s t denotes the scaled first derivative of the log density with respect to thevector f t : s t = S t ∆ t , ∆ t = ∂ log p ( y t | Y t − ) ∂f t . (4)The matrix A is a diagonal matrix of gains which are estimated via maximumlikelihood along the unknown elements of the matrix B and the initial vector f .The matrix S t is a scaling matrix set to the Moore–Penrose inverse of the expectedinformation matrix given by S t = (cid:104) diag( I t ) (cid:105) − , I t = E (cid:2) ∆ t ∆ (cid:48) t | Y t − (cid:3) . (5)Following Delle-Monache et al. (2020) and Lucas and Zhang (2016) only the diagonalelements of the information matrix are used to improve estimation.Since the observations are conditionally cross-sectionally independent the scoreand the expected information matrix can be expressed conveniently as∆ t = m (cid:88) i =1 δ i,t ∆ i,t , (6)where ∆ i,t = ∂ log p ( y i,t | Y t − ) /∂f t , while the information matrix becomesE (cid:2) ∆ t ∆ (cid:48) t | Y t − (cid:3) = m (cid:88) i =1 δ i,t E (cid:2) ∆ i,t ∆ (cid:48) i,t | Y t − (cid:3) . (7)In score driven models time-varying parameters are fully deterministic conditionalon past information. Therefore, if all series are modelled contemporaneously theGDP nowcast cannot be updated once related series become observable for the lastmonth of the quarter. To overcome this problem the related series are lead of oneperiod, as in Gorgi et al. (2019).The next subsection details the dynamics of the components in f t . Each location, scale and shape parameter is decomposed into a stochastic trendrepresenting idiosyncratic secular changes and a common component capturingcyclical variations common to all series which take the following form λ ji,t +1 = λ ji,t + A jλi s jλi,t , i = 1 , ..., m, (8)Φ j ( L ) π jt +1 = A jπ s jπ,t , (9)for all periods t = 1 , ..., T . Location parameters are indicated by j = µ , scaleparameters by j = σ and shape parameters by j = α . The information matrix needs to be inverted at each step of the recursion because its componentsare dynamic which considerably slows down estimation. To overcome this technical issue the scoredriven recursion (hence the log likelihood evaluation as well) is written in C++ (I am grateful toCaterina Schiavoni for her help with this) while the optimisation and the remainder of the analysisis carried in R. π µt , is an AR(2) process as is commonlyspecified for the business cycle; it is constrained to be stationary using the parametertransformation of Osborn (1976). The common factor in the scales captures commonvolatility shocks, like the one generated by the Covid-19 pandemic, and is specified asa stationary AR(1). The common factor in the shapes captures simultaneous shiftsin the asymmetry of prediction errors, like those typically arising at the beginningof recessions, and is also modelled as a stationary AR(1). Adding a second lag inthe common factors of scales and shapes has little effect on estimation.The stochastic trend in each parameter, λ ji,t , is useful to capture long-termdeviations. Antolin-Diaz et al. (2017) notably demonstrate that random-walkspecifications for time-varying parameters are robust to discrete breaks. They alsostress the importance of capturing secular changes in economic growth, a findingalso discussed by Doz et al. (2020).The next subsection discusses how trends and common components are relatedto the location, scale and shape parameters. Each series follows the predictive model : y i,t = µ i,t + σ i,t (cid:15) i,t , (10)for i = 1 , ..., m , t = 1 , ..., N , and where v i,t = σ i,t (cid:15) i,t is a prediction error (and (cid:15) i,t its standardised counterpart) following an arbitrary distribution with time-varyinglocation, scale and shape parameters ( µ i,t , σ i,t and α i,t ). These are decomposed intolatent states whose dynamics come from the scaled score as outlined in the previoussection. These latent states include a component common to all variables. However,the data used for estimation are of different nature; while GDP is a quarterly variable,the related series are monthly variables. It is important to account explicitly forthis mismatch in the frequency of aggregation when relating the latent states to theparameters.To address this temporal aggregation mismatch it is useful to start from theaccounting relationship between monthly and quarterly variables, specifically aquarterly variable in levels Y i,t must be equal to the three-month sum of its monthlysub-components ˜ Y i,t − i , i = 0 , , Y i,t = ˜ Y i,t + ˜ Y i,t − + ˜ Y i,t − . (11)To account for heteroskedasticity and multiplicative components, however, the dataare generally taken in logarithms, such that the variables modelled are Y ∗ i,t = log Y i,t ,and ˜ Y ∗ i,t = log ˜ Y i,t . Since the sum of the logarithms is not equal to the logarithm ofthe sum, the accounting constraint takes a nonlinear form : Y ∗ i,t = log (cid:104) exp ˜ Y ∗ i,t + exp ˜ Y ∗ i,t − + exp ˜ Y ∗ i,t − (cid:105) . (12)7his type of non-linearity becomes difficult to handle once the first differencesin logs are modelled. It is possible, however, to work from a linear approximation.Salazar et al. (1997) and Mitchell et al. (2005) show that (cid:88) i =0 h( z t − i ) ≈ (cid:32) (cid:80) i =0 z t − i (cid:33) , (13)where h ( . ) is a non-linear transformation and z t a smooth variable. The approxima-tion (13) is a second order approximation because the first order errors sum to zero.When monthly values in a quarter are close to the monthly average over the quarter,which is usually the case with seasonally adjusted figures, the approximation errorintroduced is negligible. Using this approximation equation (12) can be written as Y ∗ i,t = log3 + 13 ˜ Y ∗ i,t + 13 ˜ Y ∗ i,t − + 13 ˜ Y ∗ i,t − . (14)Using approximation (14) it is possible to write the quarter-on-quarter logdifference y i,t = Y ∗ i,t − Y ∗ i,t − as a function of the monthly log differences ˜ y i,t − i =˜ Y ∗ i,t − i − ˜ Y ∗ i,t − h − , h = 0 , ..., y t,i = 13 ˜ y i,t + 23 ˜ y i,t − + ˜ y i,t − + 23 ˜ y i,t − + 13 ˜ y i,t − . (15)Equation (15) is also discussed by Mariano and Murasawa (2003) who have popular-ised its use in mixed-frequency dynamic factor models. Location Parameters
There is a linear relationship between each variable and its location parameterwhich makes it possible to split the location of a quarterly variable into monthlycomponents and relate them to the location with (15). Importantly this step does notrequire any assumption about the distribution of the unobserved monthly variable.Monthly locations are modelled as˜ µ i,t = λ µi,t + Λ µi ( L ) π µt , (16)where the factor loading has a lag structure as in Doz et al. (2020), albeit limited toone period only, specifically Λ µ ( L ) = Λ µi, + Λ µi, L . The lagged factor loading is usedonly for series largely derived from employment data, that is the index of workinghours and the Weekly Economic Index. The factor loading is constrained to be onefor GDP.Location parameters in quarterly series are related to the monthly model usingapproximation (15) as µ i,t = 13 ˜ µ i,t + 23 ˜ µ i,t − + ˜ µ i,t − + 23 ˜ µ i,t − + 13 ˜ µ i,t − , (17)while for monthly series it is simply µ i,t = ˜ µ i,t . (18)8 cale and Shape Parameters Diverging from the normal distribution is useful to capture the features of the datain a flexible way, but it also creates challenges in a mixed-frequency framework.Notably, while location parameters can be disaggregated temporally without makingany statistical assumptions on the monthly sub-components, that is not possiblefor scale and shape parameters. This is problematic because the sum of two non-normally distributed variables with known distributions has a distribution which isdifficult to evaluate and in many cases unknown. Linking the monthly model to thequarterly model is therefore difficult.When working in a non-Gaussian framework it is preferable to model the de-pendencies across series in scale and shape parameters using quarterly models. Todo so related series exhibiting common factors in their scale and shape parametersare aggregated into rolling quarterly observations (quarter-on-quarter deviationsobserved monthly). This introduces serial correlation in these series which is ad-dressed by modelling monthly location components, as outlined above. The temporalaggregation strategies for modelling jointly quarterly and monthly observations andmodelling rolling quarterly observations (quarterly observations observed monthly)are essentially the same; in both cases it is necessary to go back to the underlyingmonthly model. Labonne and Weale (2020) show that when rolling quarterly seriesare not subject to important measurement errors it is possible to interpolate themonthly path very precisely with (14). Hence there should be little loss of informationwhen using rolling quarterly observations instead of monthly observations.Separately, while locations can take any real value, scale parameters are positiveand shape parameters can take values only in (0;1) in the distribution used below.To incorporate these constraints during estimation, trends and common factors arerelated to scale and shape parameters as σ i,t = exp( λ σi,t + Λ σi φ σt ) , α i,t = 1 / (1 + exp( λ αi,t + Λ αi φ αt )) , (19)with the factor loadings constrained to one for the GDP series ( i = 1). The Special Case of a Normal Distribution
In a Gaussian setting it is possible to link monthly variances to quarterly variancesusing an approximation close to (15) as shown below.If y i,t is conditionally normally distributed, its conditional variance is equal tothe variance of the prediction error such that V ar ( y i,t | Y t − ) = E [( y i,t − E ( y i,t | Y t − )) ] = E ( v i,t ) = V ar ( v i,t ) . Assuming that series i is observable quarterly, the quarterly prediction error can besplit into its monthly components using approximation (15) as v i,t = 13 ˜ v i,t + 23 ˜ v i,t − + ˜ v i,t − + 23 ˜ v i,t − + 13 ˜ v i,t − , v i,t is a monthly prediction error. Using this approximation the variance ofthe prediction error can be decomposed into monthly variances as V ar ( v i,t ) = 19 V ar (˜ v i,t ) + 49 V ar (˜ v i,t − ) + V ar (˜ v i,t − ) + 49 V ar (˜ v i,t − ) + 19 V ar (˜ v i,t − ) , (20)since, assuming that the monthly errors ˜ v i,t are i.i.d, the covariance terms are zero.This step is possible only if the prediction errors are normally distributed. Therolling quarterly scale parameter σ i,t can thus be written as a function of the monthlyscale parameter ˜ σ i,t as σ i,t = (cid:114)
19 ˜ σ i,t + 49 ˜ σ i,t − + ˜ σ i,t − + 49 ˜ σ i,t − + 19 ˜ σ i,t − . (21)It is thus possible to specify a monthly model for all scale parameters where themonthly components are related to the quarterly scale parameter of GDP using (21)with ˜ σ ,t = exp( λ σ ,t + Λ σ φ σt ). Each series’ conditional density comes from the family of asymmetric student-t(AST) density of Zhu and Galbraith (2010). The distinctive features of the ASTare its shape parameter, or skewness parameter, which controls the asymmetryround the central part of the distribution, and its tail parameters which control tailbehaviours independently on each side.The log AST density of observation t of series i takes the formlog p i ( y i,t | Y t − ) = − ln σ i,t − ν ,i + 12 ln (cid:104) ν ,i (cid:16) y i,t − µ i,t α i,t σ i,t K ( ν ,i ) (cid:17) (cid:105) y i,t ≤ µ i,t ) − ν ,i + 12 ln (cid:104) ν ,i (cid:16) y i,t − µ i,t − α i,t ) σ i,t K ( ν ,i ) (cid:17) (cid:105) y i,t > µ i,t ) , (22)where σ i,t is the scale parameter, α i,t is the shape parameter which can take values in(0 , ν ,i and ν ,i are respectively the left and right tail parameters which takepositive values. K ( ν ) = Γ(( ν + 1) / / ( √ νπ Γ( ν/ . ) is the Gamma function)and 1( x ) is an indicator variable equal to one if statement x is true and zero otherwise.The distribution is skewed towards positive values if α i,t < . α i,t > .
5. When the tail parameters are constrained to be very largeand the skewness parameter to 0.5 the AST is equivalent to a (scaled) normaldistribution.Figure 1 illustrates the effects that scale, shape and tail parameters have on thedensity function. The solid red line shows the AST density when σ = 1 while otherparameters are constrained to be Gaussian ( α = 0 . ν = ν = ∞ ). The dottedblue line shows the AST density when either the scale, tail or shape parametersare varying. The density at the location changes when the scale parameter is10ltered but is independent to variations in the tail and shape parameters. Thisis a particular feature of this version of the AST distribution (given in equation(5) of Zhu and Galbraith (2010)) where the random variable is scaled with B i,t = α i,t K ( ν i, ) + (1 − α i,t ) K ( ν i, ). −2 −1 0 1 2 . . . (a) Normal with standard error x 2 Prediction error −2 −1 0 1 2 . . . (b) AST with tail parameters = 3 Prediction error D en s i t y −2 −1 0 1 2 . . . (c) AST with left tail parameter = 3 −2 −1 0 1 2 . . . (d) AST with shape parameter = 0.8 D en s i t y Figure 1: Illustration of the density functions from AST distributions with differentset of parameters. With shape parameter = 0.5 and tail parameters = ∞ the ASTdistribution reduces to a (scaled) normal distribution (red solid line).Graph (a) shows the effect of increasing the scale parameter, specifically to σ = 2.The dispersion increases symmetrically on both sides and can be interpreted as ageneral (or symmetric) increase in forecasting uncertainty when the model is appliedto conditional GDP growth.Graph (b) shows the effect of lowering both tail parameters to three. Theprobability in the tails increases which can be used to account for extreme events.Graph (c), on the other hand, illustrates asymmetric tails; specifically an heavy tailon the left side and Gaussian tail on the right side ( ν = 3 while ν = ∞ ). Thisincreases the probability of extreme negative events while leaving the probabilityof large positive extreme events unaffected. This asymmetric tail behaviour canbe particularly useful for conditional GDP if, as Jensen et al. (2020) argue, thebusiness cycle has become more asymmetric in the last thirty years. They find thatcontractions have become more violent while recoveries remain smooth.Finally, graph (d) shows the effect of negative skewness ( α = 0 .
8) on the densityfunction. The central part of the density becomes skewed heavily toward negativevalues. This statistical behaviour should be especially useful to capture the onsetof recessions when coupled with an increase in dispersion : while general economicuncertainty increases, the likelihood of a positive outcome decreases. The real-timeanalysis of US GDP in the first quarter of 2020 shown in section 7 validates thisintuition.Figure 2 illustrates the response function of the scaled score for location, scaleand shape parameters with the prediction error as input. These are useful for The components of the score and information matrix are shown in appendix A −2 −1 0 1 2 . . . Density −2 −1 0 1 2 − Scaled score (location) S c a l ed sc o r e −2 −1 0 1 2 Scaled score (scale)
Prediction error −2 −1 0 1 2 − − Scaled score (shape)
Prediction error S c a l ed sc o r e Gaussian model Fat tailed model (tail parameters = 3)Fat tailed model (tail parameters = 30) Skewed model (shape parameter = 0.8)
Figure 2: Plots of the density and score functions for an asymmetric student-t (AST)distribution with σ = 1. X-axes show prediction errors. With shape parameter α = 0 . ν = ν = ∞ the AST is equivalent to a (scaled) normaldistribution. On the importance of high tail parameters for identifying turning points
Downweighting the effect of large prediction errors is usually a desirable feature ofscore driven models because it leads to a more robust estimation. However, economiccrises, and the Covid-19 period in particular, are examples of cases where outliersmost likely have important and long-lasting effects on means and variances of timeseries. If the related series’ distributions are allowed to have fat tails, the effect oflarge swings in the economic activity on the time-varying parameters are likely tobe downweighted. Consequently the model might not capture turning points in theeconomic activity and the concurrent increase in the dispersion of prediction errorsin a timely way.To investigate thoroughly the effects of fat tails on estimation and forecast-ing performance in recessionary episodes the empirical analysis compares modelswith unconstrained tail parameters with models constrained to have Gaussian tailparameters in the related series. 12
A Weighted Maximum Likelihood Estimator
The estimation strategy relies on the weighted maximum likelihood method ofBlasques et al. (2016), notably applied in a score driven framework by Gorgi et al.(2019). It accounts explicitly for the fact that, while related series are used forestimation, the primary objective here is to forecast GDP growth, not the relatedseries; these are used solely to improve GDP nowcasts.To see this more clearly it is helpful to look at the model’s log likelihood atperiod t , given by (2), which is the sum of each observation’s log likelihood at time t . In a misspecified setting the parameters that maximise the total log likelihoodare not necessarily those that maximise the log likelihood of GDP. This issue iseven more prominent in a mixed-frequency framework where GDP is observed onceevery three months whereas the related series are observed monthly. Indeed whenan observation is missing the series it relates to has no impact on the model’s loglikelihood for this period. The log likelihood associated to GDP has consequentlyless weight on the total log likelihood than those of the related series.Using the weighted maximum likelihood approach the vector of unknown para-meters Θ which includes the distributions’ parameters, the initial values of thetime-varying parameters, the autoregressive coefficients and the gains is estimatedas ˆΘ = arg max Θ N (cid:88) t =1 log p w ( y t | Y t − ) , (23)where log p w ( y t | Y t − ) is the weighted log likelihood defined aslog p w ( y t | Y t − ) = δ i,t log p ( y ,t | Y t − ) + W m (cid:88) i =2 δ i,t log p i ( y i,t | Y t − ) . (24)The weight W is applied to the related series’ log likelihood and diminishtheir contribution to the total log likelihood. It cannot be estimated alongsidethe other parameters and Blasques et al. (2016) suggest selecting it via cross-validation techniques. Alternatively, Gorgi et al. (2019) set the weight to zero.This complicates the identification of the scale parameters in the related series; aproblem they overcome by modelling a unique scale parameter for all series. Thiswould be problematic here since the modelling approach proposed relies on a flexiblespecification for scale parameters. Furthermore, in their out-of-sample nowcastingexercise Gorgi et al. (2019) do not find that the score driven approach yields a clearbenefit when the contribution of the indicator series is null.Here the weighting factor is set to one third to compensate for the implicitdownweighting of the GDP series coming from its lower frequency of observationcompared to the related series. The indicator series’ log likelihood contributions arenot downweighted further because that might deteriorate excessively the model’scapacity to predict related series in real time. This could yield to prediction errorsloosing their economic meaning and with it their ability to indicate period ofeconomic depressions and uncertainty, which is the focus of this paper.13 Real-Time Data on US Economic Growth
This study is centred on quarterly GDP which is the leading measure of economicgrowth and the data are taken from the United States. The estimate of GDPanalysed is the
Advance Estimate , the timeliness estimate of GDP in the US.Three monthly series related to economic growth are used to improve GDP now-casts: Industrial production, the index of working hours and the Weekly EconomicIndex (Lewis et al. (2020)). Industrial production is a component of GDP but ismore frequent and timely. The index of working hours is a very timely indicator ofemployment and notably reflected the effect of the Covid-19 restrictions from theonset of the pandemic. The Weekly Economic Indicator has been created specificallyto capture the effect of the pandemic in US in a timely way.GDP and industrial production are regularly subject to benchmark changeswhich affect the entire series, such as changes in indexation. To deal with thesechanges all series are taken in first differences in logs. The sample includes datafrom January 1973 up to June 2020 for GDP, industrial production and the indexof weekly hours whereas the Weekly Economic Index is used from June 2010 up toJune 2020. The series are illustrated in figure 3 using the vintage available at theend of July 2020. −15−10−505 1980 1990 2000 2010 2020(a) GDP (b) IP (c) Index of Working Hours (d) Weekly Economic Index
Figure 3: Quarterly data (calendar quarters) from January 1973 up to June 2020.Industrial production, the index of working hours and the Weekly Economic Indexare aggregated into quarterly figures for comparison with GDP. Data from the mostrecent vintage. Shaded areas indicate the US recessions classified by the NBER.GDP, industrial production and the index of working hours are regularly revisedover time because the data used for their compilation accrue gradually. Therefore,to investigate the forecasting performance of the model in real time it is necessary toestimate the model recursively using successive vintages of the data. Such vintagesare produced by the Federal Reserve Bank of Philadelphia. The Weekly Economic14ndicator, on the other hand, is revised only marginally.The most timely series are the index of working hours and the Weekly EconomicIndex which are released at the beginning of each month and relate to the previousmonth. Industrial production is released in the middle of each month and relate to theprevious month while GDP is released at the end of the month following the quarterit relates to. Hence, two series (the index of working hours, the Weekly EconomicIndex) are approximately released simultaneously while industrial production andGDP follow different publication schedules. This yields seven rounds of revisions inbetween GDP releases, that is seven nowcasting steps.The first nowcasting step is at the end of the first month in the quarter, whenthe previous-quarter GDP figure is released. The second step is at the beginningof second month in the quarter when monthly figures for the index of workinghours and the weekly economic index are released. The next five steps are eachapproximately two-weeks apart, when figures for either industrial production or theindex of working hours and the Weekly Economic Index are released.
The primary objective of this paper is to investigate the potential gains of modellingcommon factors in scale and shape parameters on dynamic factor models’ nowcastingperformance. For this it is useful to compare a range of model specifications.The most flexible specification is model (a) which is an asymmetric student-tdynamic factor model (DFM) with common factors in location, scale and shapeparameters. While GDP follows an unrestricted AST distribution, IP and IWH areconstrained to have a unique tail parameter and effectively follow skewed Student-tdistributions. Model (b) is derived from model (a) but tail parameters in the relatedseries are constrained to be Gaussian; the related series consequently have skewednormal distributions. This means that more attention is placed on sharp movementsin the related series.Separately, model (c) is Gaussian DFM with stochastic volatilities (SV) and acommon factor in scale parameters. Since model (c) is Gaussian, the common scalecomponent can be modelled at a monthly frequency and related to the quarterlyscale in GDP using (21). Comparing model (c) to the non-Gaussian models (a)and (b) is useful to understand if the gains from modelling non-Gaussian featuresoutweigh the loss in precision resulting from the necessary aggregation of the relatedseries featuring scale and shape common factors.These three models, which are designed to make full use of the cross-sectionalinformation in the data, are compared to benchmark models which do not featurescale and shape common factors. Model (d) is an asymmetric Student-t modelcomparable to model (b) but without scale and shape common factors. Thisspecification is closer to the non-Gaussian models of Delle-Monache et al. (2020)(their model is a univariate model with many regressors to improve estimation oftime-varying parameters whereas model (d) is a dynamic factor model).15pecification (e) is a Gaussian DFM with stochastic volatilities which is a proxyfor current dynamic factor models. It is derived from model (c) but does not featurea common scale component. The relative performance of model (c) compared tomodel (e) provides an indication of the gains stemming from augmenting dynamicfactor models with a scale common factor.Finally, model (f) extends model (e) to Student-t distributions. Comparingmodel (e) and (f) is useful for understanding the effect that low tail parameters mayhave on the ability of the model to capture turning points.All specifications except (a) and (b) . The Weekly Economic Index, on theother hand, has a normal distribution with constant scale parameter because ofthe short sample of available observations. Consequently WEI remains monthlywhich is a common feature in all models. A table summarising the different modelspecifications is shown in appendix B.
First the models are analysed using the vintage available at the end of July 2020,the latest available vintage in this study.
Modelling common scale and shape components significantly improvesin-sample fit
The first two columns of Table 1 show the log likelihood of each model and the loglikelihood of the GDP series only, while the third and fourth columns show the AICand BIC information criteria. The total log likelihood and information criteria ofthe first two models ((a) and (b)) cannot be compared to the other models becausethey are estimated with different data. For these two models industrial productionand the index of working hours are rolling quarterly whereas they are monthly inthe other models.Table 1: Comparison of the model specifications using the full-sample results.
Model L.L. L.L. GDP AIC BIC(a): AST DFM with Scale+Shape CFs -381.4 -124 854.7 1097.5(b): AST DFM (Sk Covs) with Scale+Shape CFs -402.3 -124.6 892.7 1124.9(c): DFM with SV and Scale CF -548.9 -135.9 1159.8 1323.4(d): AST DFM (Sk Covs) with SV and TV shape -578.1 -136.3 1220.2 1389.1(e): DFM with SV -580.1 -138 1212.1 1349.4(f): St-t DFM with SV -569 -139.4 1196 1349.1Note: L.L. refers to the model’s total log likelihood. L.L. GDP refers to the log likelihoodof the GDP series only. AIC = − × Log Lik. + 2 × p ; BIC = − × Log Lik. + log N × p ; N is the number of observations and p the number of parameters estimated. The results from model (a), (b) and (c) show that modelling common factorsin scale and shape parameters yields to large improvements in the log likelihood16f GDP, with a particularly large benefit when modelling a shape common factor.Separately, the Gaussian model with scale common factor (model (c)) does better,in all metrics, than the benchmark models which do not feature a scale commonfactor. This highlights the importance of capturing cross-sectional dependencies inscale parameters.
Modelling fat tails does not necessarily improve the log likelihood ofGDP
Modelling covariates with fat tails in the AST model improves the total log likelihood,the log likelihood of GDP (albeit marginally) and the information criteria. But whilemodelling fat tails in a dynamic factor model with stochastic volatility (specification(f)) improves the total log likelihood and information criteria compared to a modelfeaturing Gaussian series, the likelihood of GDP deteriorates.The decline in the predictive capacity of GDP when related series are allowed tohave fat tails can be explained from the downweighting of large prediction errorsin the related by the score functions. These prediction errors provide a signal onunforeseen turning points in the economic activity, and low tail parameters preventthis signal to be propagated to the GDP predictions. This result is consistent withthe real-time analysis below which shows that fat-tails in the related series increaseGDP nowcasting performance in normal times but reduce it in recessions.
The common scale component provides a consistent historical picture offorecasting uncertainty
Figure 4 shows the estimated common scale components. All three models featuringa common scale component yield broadly similar estimates. First, they reflect anoverall decrease in volatility starting in the second half of the eighties which hasbeen documented extensively in the macroeconomics literature since McConnelland Perez-Quiros (2000). Secondly, they feature sharp spikes during the recessionstriggered by the 2007 financial crisis and lately by the coronavirus pandemic. Whilethe common scale component at the time of the financial crisis reaches levels notseen since the seventies and early eighties, its level in the first two quarters of2020 is unprecedented. This reflects the stark forecasting uncertainty generatedby the government lockdown policies in response to the pandemic. Overall thecommon volatility factor carries economic meaning and gives a consistent picture offorecasting uncertainty historically and across different model specifications.
Recessions are generally associated with rising negative skewness
Location, scale and shape parameters derived from all models are compared in Figure(5). Unlike scale and shape parameters, location parameters show little divergenceacross models except at the time of the Covid-19 pandemic. All models feature adecrease in forecasting uncertainty starting in the mid-eighties. Interestingly the17
123 1980 1990 2000 2010 2020(a): AST DFM with Scale+Shape CFs (b): AST DFM (Sk Covs) with Scale+Shape CFs(c): DFM with SV and Scale CF
Figure 4: Scale common factors. Rolling quarterly estimate for model (a) and(b) and monthly estimate for model (c). Shaded areas indicate the US recessionsclassified by the NBER.. DFM = Dynamic Factor Model. SV = Stochastic Volatility;CF = Common Factor.increase in forecasting uncertainty in recessions is generally associated with a sharpincrease in the skewness parameter: the probability of negative prediction errorsincreases while positive prediction errors become less likely.The results from the model with dynamic shape components are consistentwith the findings of Carriero et al. (2020) and Delle-Monache et al. (2020) whoshow that, while statistical evidence for skewness in GDP growth is generally weak,this masks an erratic behaviour with periods of positive and negative skewness.While skewness fluctuates round zero until the mid-eighties, it becomes increasinglynegative afterwards when overall volatility decreases. The financial crisis marks achange with a period of positive skewness.Analysing the different model specifications using the entire sample is useful tounderstand the statistical features of the models but cannot be used to infer theireffectiveness in real time. The next section shows the results from a recursiveestimation simulating a real-time environment.
In this section the models are estimated recursively over a 20-year period from June2000 up to June 2020 using historical vintages. Thus it is possible to analyse theirperformances in pseudo real-time. It is not in exactly in real-time because the analysis was carried in August 2020 when all GDPfigures were already released. hapeScaleLocation1980 1990 2000 2010 2020−10.0−7.5−5.0−2.50.02.505100.30.40.50.60.70.8 (a): AST DFM with Scale+Shape CFs (b): AST DFM (Sk Covs) with Scale+Shape CFs(c): DFM with SV and Scale CF (d): AST DFM (Sk Covs) with SV and TV shape(e): DFM with SV (f): St−t DFM with SV Figure 5: Quarterly figures corresponding to calendar quarters. Shaded areasindicate the US recessions classified by the NBER. SV = Stochastic Volatility; CF= Common Factor.
Deriving multi-step ahead density nowcasts
The GDP density nowcast in the last step of the nowcasting window is directlygiven by the one-step ahead prediction error density of GDP. But for earlier stepsit is important to account for the uncertainty induced by missing values in therelated series. This is done by drawing vectors of observations for each period with19issing observations and using the score driven recursion to retrieve the time-varyingparameters in the next period for each draw, which are then used to generate newobservations. Specifically, if the target is the GDP nowcast in period t , but relatedseries are missing from t −
2, then the one-step ahead densities of the related seriesat t − t −
2. Whenobservations in related series are only missing partially the draws are replaced bythe observed prediction errors for the related series which have been released. Thescore driven recursion is used on each vector of prediction errors which yields sets ofscale, shape and location parameters for period t −
1. These new sets of scale andshape parameters yield new sets of one-step ahead densities which are used to drawvectors of prediction errors round the vector of locations in t −
1. Eventually, eachvector of prediction errors in t − t through the scoredriven recursion (related series are modelled with a lead of one period comparedto GDP). The density nowcast is given by the empirical density attached to theseGDP nowcasts. Average Log Score
The accuracy of probabilistic predictions, or density nowcasts, is analysed using theaverage log score at each nowcasting step. The log score of a given nowcast is givenby the nowcasted log density at the observed value. The better the probabilisticforecast, the higher the log score.Forecasters and policy makers are particularly interested in the performanceof forecasting methods in times of economic troubles. To address this questionthe results are analysed over five samples which cover the Covid-19 pandemic, thegreat financial crisis (GFC), all recessions taken from the NBER classifications since2000, normal times (all periods except recessions) and finally a sample covering allquarters.As more data accrue during the nowcasting quarter the average log score shouldincrease monotonically. However, while the average log score is generally highertowards the end of the nowcasting window, it does not exhibit a monotonic improve-ment. This can be explained by the relatively low numbers of series modelled andthe resulting low numbers of release dates during the quarters. Nevertheless thisstatistic remains useful to compare the models’ predictive capacities.First, scale and shape common factors improve the forecasting performanceduring recessions with a particularly large benefit at the end of the nowcastingwindow. Up until the last nowcasting step the Gaussian DFM with scale commonfactor performs especially well. The common factor in the scales captures sharpincreases in forecasting uncertainty in a timely way, and unlike with the AST modelsfeaturing both scale and shape common factors, all related series can remain monthly;hence there is no loss in precision at the beginning of the nowcasting window. At theend of the nowcasting window, however, there is a benefit in modelling non-Gaussian20 ecessionsGFC Normal timesAll quarters Covid−191234567 1234567−20−10−0.60−0.55−0.50−0.45−1.0−0.8−0.6−1.2−1.0−0.8−0.6−5−4−3−2−1 (a): AST DFM with Scale+Shape CFs(b): AST DFM (Sk Covs) with Scale+Shape CFs(c): DFM with Scale CF(d): AST DFM (Sk Covs) with SV and TV shape(e): DFM with SV(f): St−t DFM with SV
Figure 6: Average log score. Higher values imply better density forecasts. x-axesshow the numbers of steps towards the release of the GDP number in the nowcastingwindow. New data are released at each step. DFM = Dynamic Factor Model. SV =Stochastic Volatility; CF = Common Factor. Covs = Covariates.features and a common factor in the shape parameters in particular.Both Gaussian models yield relatively similar performances up until the lastnowcasting step, where the benefit of modelling a scale common factor clearly givesan advantage. The uncertainty attached to the GDP nowcast in both models canbe updated through the dispersion of the common factor in the locations. But thismechanism takes place with a lag of one period because the dispersion of the locationcommon factor is derived from the expected dispersion of the related series’ forecasterrors. Therefore, the uncertainty attached to the GDP nowcast is not updated onceall related series have been observed in the quarter, unless there is a common factorin the scale parameters.Separately, in the first half of the nowcasting window the AST models with scaleand shape common factors (model (a) and (b)) perform less well than the othermodels excluding the Student-t DFM model (f). This most likely represents thecost of aggregating the related series featuring scale and shape common factorsinto rolling quarterly figures. However, AST models with scale and shape commonfactors do consistently better at the end of the nowcasting window.The Student-t model performs especially poorly during recessions. This stemsfrom the model’s inability to capture turning points in both the location and scaleparameters due to the downweighting of prediction errors by the score function when21ail parameters are low, as illustrated in figure 2. This downweighting is subdued inthe AST model (a) because the related series feature larger tail parameters whenaggregated quarterly. In normal times, however, this downweighting is beneficialbecause the central part of GDP density nowcasts is not affected by relatively largeerrors in the related series. In fact the Student-t DFM model does consistentlybetter in normal times.Figure 7 makes it easier to compare the accuracy of density nowcasts acrossmodels when considering both recessions and normal times. It shows the average logscore at each nowcasting step for each model. The darker the colour, the higher logscore and thus the better the density nowcast. Model (c), the Gaussian DFM withscale common factor, is the only model to perform well during the entire nowcastingwindow. It is only out-performed by the AST models with scale and shape commonfactors in the last nowcasting step.Figure 7: Average log score across all periods at each nowcasting step. The lowerthe nowcasting step the closer the release. The higher the log score the better thedensity forecast. And the darker the background colour the higher the log score.
To understand why models with a scale common factor or both scale and shapecommon factors do better as more data accrue during the quarter and the GDPrelease gets closer, it is useful to analyse a period of stark economic uncertainty andthe first two quarters of 2020 provide a good illustration for this.Figure 8 shows the density nowcasts for the first quarter of 2020 starting fromthe March release of the industrial production figure relating to February. The effectof Covid-19 started to appear in the related series in March, and it was clear at thetime that the economic environment was growing more uncertain. But the modelswith cross-sectionally independent stochastic volatilities (model (d) and (e) and (f))show a relatively stable nowcasting uncertainty. Although this is counter-intuitive itis an inevitable feature of these models.In normal times, the uncertainty attached to a nowcast should decline graduallyas more data accrue during the quarter. This is because the uncertainty attachedto each coming release is alleviated once the figures are published. In other words,the forecast horizon of the related series decreases after each release, and since22 . . . . (a): AST DFM with Scale+Shape CFs Q/Q growth rateMid MarchEarly AprMid Apr −3 −2 −1 0 1 2 3 . . . . (b): AST DFM (Sk Covs) with Scale+Shape CFs Q/Q growth rate den s i t y _ r e s u l t s _a ll $ V C F [[ ]] $ y −3 −2 −1 0 1 2 3 . . . . (c): DFM with SV and Scale CF Q/Q growth rate −3 −2 −1 0 1 2 3 . . . . (d): AST DFM (Sk Covs) with SV and TV shape Q/Q growth rate den s i t y _ r e s u l t s _a ll $ SV [[ ]] $ y −3 −2 −1 0 1 2 3 . . . . −3 −2 −1 0 1 2 3 . . . . (e): DFM with SV Q/Q growth rate−3 −2 −1 0 1 2 3 . . . . −3 −2 −1 0 1 2 3 . . . . (f): St−t DFM with SV Q/Q growth rate den s i t y _ r e s u l t s _a ll $ G _ FT [[ ]] $ y −3 −2 −1 0 1 2 3 . . . . Figure 8: Real-time density nowcasts of US GDP Q1. The dashed line indicates thepublished figure. DFM = Dynamic Factor Model. SV = Stochastic Volatility; CF =Common Factor. TV = Time Varying.the scale parameters remains relatively stable in normal times (or changes verymoderately due to the expanding sample), forecast uncertainty decreases naturally. Ashortcoming of models with constant variance parameters is that this reasoning alsoapplies to periods of economic stress. As the forecast horizon decreases, nowcastinguncertainty can only decrease, and this even if the economic environment becomesmore uncertainty.Economic uncertainty is generally signalled by increasing prediction errors inseries related to economic growth. By modelling stochastic scale parameters in theserelated series, larger prediction errors can be translated into greater conditionalvariances (the one-step ahead forecast error variances) in the related series.Importantly, changes in the related series’ forecast uncertainty also affect theuncertainty attached to the GDP nowcast, albeit with a lag. A greater forecastuncertainty in the related series, or more precisely an increase in the one-step aheadforecast error variances, increases the dispersion of the two-step ahead estimate ofthe common location component. Since the common location component is shared byall series, the dispersion in the GDP forecast increases as well. Hence, even thoughthe scale parameters in stochastic volatility models are contemporaneously cross-sectionally independent, forecast error variances are related through the location23ommon factor.But there is a lag of a one period before forecast errors in the related series affectthe forecast error variance of GDP; the uncertainty attached to the one-step aheadGDP forecast is not affected. The progress of the Covid-19 pandemic in the USprovides an important illustration of this fact.Modelling a common factor in the scale parameter alleviates this problem becausethe one-step ahead forecast error variances are now related directly. In the Gaussianmodel with scale common factor and the AST models with scale and shape commonfactors the uncertainty attached to the GDP nowcast for March increases graduallyfollowing the release of the related series. This increase in uncertainty is necessaryto generate realistic probabilistic forecasts for March.Moreover, the AST models with a shape common component can capture theasymmetry in the uncertainty attached to the nowcast in addition to a greaterdispersion. Negative prediction errors are significantly more likely than positiveones. However, the AST model with fat tails (model (a)) produces a poorer densitynowcast compared to the AST model with Gaussian tail parameters in the relatedseries. Low tail parameters downweight the signal from the related series’ predictionerrors; hence the location, scale and shape parameters do not adjust enough to yielda satisfactory density nowcast.The analysis of the June nowcast shown in figure 9 is more complicated because ofthe erratic behaviour of the projections in this quarter coming from the unprecedenteddownturn in the activity and resulting magnitude of the prediction errors in therelated series. However, it is the AST model with scale and shape common factorand Gaussian tail parameters in the related series (model (b)) that yields the bestprobabilistic forecast at the end of the nowcasting window. Again the Gaussianmodel with common volatility component and the AST models with scale and shapecommon factors capture the uncertainty arising late in the nowcasting window whichthe other models do not identify.
This paper has shown how common components in scale and shape parameters maybe modelled in Gaussian (scale only) and non-Gaussian mixed-frequency dynamicfactor models. The empirical application in pseudo real time using US economicgrowth data shows that modelling scale and shape common factors yields betterdensity nowcasts towards the end of the nowcasting window in recessionary episodes.While this paper uses a score driven approach for estimation, the gains in real-timeperformance stemming from modelling a common volatility factor should arise inother modelling strategies such as mixed-frequency vector auto-regressions and statespace models which allow for stochastic volatility specifications. This is not necessarily true for models where unobserved states have their own sources oferrors. But modelling the dependencies in conditional variances directly though scale parametersshould nevertheless improve nowcast uncertainty in these models.
20 −15 −10 −5 0 5 . . . . (a): AST DFM with Scale+Shape CFs Q/Q growth rateLate AprilEarly MayMid MayEarly JuneMid JuneEarly JulyMid July −20 −15 −10 −5 0 5 . . . . (b): AST DFM (Sk Covs) with Scale+Shape CFs Q/Q growth rate den s i t y _ r e s u l t s _a ll $ V C F [[ ]] $ y −20 −15 −10 −5 0 5 . . . . (c): DFM with SV and Scale CF Q/Q growth rate −20 −15 −10 −5 0 5 . . . . (d): AST DFM (Sk Covs) with SV and TV shape Q/Q growth rate den s i t y _ r e s u l t s _a ll $ SV [[ ]] $ y −20 −15 −10 −5 0 5 . . . . −20 −15 −10 −5 0 5 . . . . (e): DFM with SV Q/Q growth rate−20 −15 −10 −5 0 5 . . . . −20 −15 −10 −5 0 5 . . . . (f): St−t DFM with SV Q/Q growth rate den s i t y _ r e s u l t s _a ll $ G _ FT [[ ]] $ y −20 −15 −10 −5 0 5 . . . . Figure 9: Real-time density nowcasts of US GDP Q1. The dashed line indicates thepublished figure. DFM = Dynamic Factor Model. SV = Stochastic Volatility; CF =Common Factor. TV = Time Varying.
References
Adrian, T., N. Boyarchenko, and D. Giannone (2019, April). Vulnerable growth.
American Economic Review 109 (4), 1263–1289.Antolin-Diaz, J., T. Drechsel, and I. Petrella (2017, May). Tracking the slowdownin long-run gdp growth.
The Review of Economics and Statistics 99 (2), 343–356.Antolin-Diaz, J., T. Drechsel, and I. Petrella (2020). Advances in NowcastingEconomic Activity:Secular Trends, Large Shocks and New Data.Banbura, M., D. Giannone, M. Modugno, and L. Reichlin (2013). Now-Castingand the Real-Time Data Flow. In G. Elliott, C. Granger, and A. Timmermann(Eds.),
Handbook of Economic Forecasting , Volume 2 of
Handbook of EconomicForecasting , Chapter 0, pp. 195–237. Elsevier.Blasques, F., S. Koopman, M. Mallee, and Z. Zhang (2016). Weighted maximumlikelihood for dynamic factor analysis and forecasting with mixed frequency data.
Journal of Econometrics 193 (2), 405–417.25arriero, A., T. E. Clark, and M. Marcellino (2016). Common drifting volatility inlarge bayesian vars.
Journal of Business & Economic Statistics 34 (3), 375–390.Carriero, A., T. E. Clark, and M. Marcellino (2018). Measuring uncertainty and itsimpact on the economy.
The Review of Economics and Statistics 100 (5), 799–815.Carriero, A., T. E. Clark, and M. Marcellino (2020). Assessing internationalcommonality in macroeconomic uncertainty and its effects.
Journal of AppliedEconometrics 35 (3), 273–293.Creal, D., S. J. Koopman, and A. Lucas (2013). Generalized autoregressive scoremodels with applications.
Journal of Applied Econometrics 28 (5), 777–795.Creal, D., B. Schwaab, S. J. Koopman, and A. Lucas (2014). Observation-drivenmixed-measurement dynamic factor models with an application to credit risk.
Review of Economics and Statistics 96 (5), 898–915.Delle-Monache, D., A. De-Polis, and I. Petrella (2020). Modelling and ForecastingMacroeconomic Downside Risk.
Economic Modelling and Forecasting Group (34).Delle-Monache, D. and I. Petrella (2017). Adaptive models and heavy tails with anapplication to inflation forecasting.
International Journal of Forecasting 33 (2),482–501.Doz, C., L. Ferrara, and P.-A. Pionnier (2020, January). Business cycle dynamicsafter the great recession: An extended markov-switching dynamic factor model.(2020/01).Durbin, J. and S. J. Koopman (2012).
Time series analysis by state space methods .Oxford University Press.Gorgi, P., S. J. Koopman, and M. Li (2019). Forecasting economic time series usingscore-driven dynamic models with mixed-data sampling.
International Journal ofForecasting 35 (4), 1735–1747.Haldane, A. (2012). Tails of the unexpected.
Paper delivered at a conferencesponsored by the University of Edinburgh Business School, Edinburgh, June 8 .Harvey, A. C. (1989).
Forecasting, structural time series models and the Kalmanfilter . Cambridge University Press.Harvey, A. C. (2013).
Dynamic Models for Volatility and Heavy Tails: With Applica-tions to Financial and Economic Time Series (Econometric Society Monographs) .Cambridge University Press.Huber, F. (2016). Density forecasting using bayesian global vector autoregressionswith stochastic volatility.
International Journal of Forecasting 32 (3), 818–837.26ensen, H., I. Petrella, S. H. Ravn, and E. Santoro (2020). Leverage and deepeningbusiness-cycle skewness.
American Economic Journal: Macroeconomics 12 (1),245–281.Labonne, P. and M. Weale (2020). Temporal disaggregation of overlapping noisyquarterly data: estimation of monthly output from uk value-added tax data.
Journal of the Royal Statistical Society: Series A (Statistics in Society) 183 (3),1211–1230.Lewis, D. J., K. Mertens, and J. H. Stock (2020). U.S. Economic Activity During theEarly Weeks of the SARS-Cov-2 Outbreak.
Federal Reserve Bank of Dallas (2011).Lucas, A. and Z. Zhang (2016). Score-driven exponentially weighted moving averagesand value-at-risk forecasting.
International Journal of Forecasting 32 (2), 293–302.Mariano, R. S. and Y. Murasawa (2003). A new coincident index of business cyclesbased on monthly and quarterly series.
Journal of Applied Econometrics 18 (4),427–443.McConnell, M. M. and G. Perez-Quiros (2000). Output fluctuations in the unitedstates: What has changed since the early 1980’s?
American Economic Re-view 90 (5), 1464–1476.Mitchell, J., R. J. Smith, M. R. Weale, S. Wright, and E. L. Salazar (2005). Anindicator of monthly gdp and an early estimate of quarterly gdp growth.
TheEconomic Journal 115 (501), 108–129.Osborn, D. R. (1976). Maximum likelihood estimation of moving average processes.
Annals of Economic and Social Measurement 5 (1), 75–87.Salazar, E., R. Smith, M. Weale, and S. Wright (1997). A monthly indicator of gdp.
National Institute Economic Review 161 (1), 84–89.Zhu, D. and J. W. Galbraith (2010). A generalized asymmetric student- distributionwith application to financial econometrics.
Journal of Econometrics 157 (2),297–305. 27 ppendix A Elements of the score and information matrix
The score with respect to location parameters is∆ µi,t = ∂ log p ( y i,t | Y t − ) ∂µ i,t = ν i, + 11 + ν i, (cid:16) y i,t − µ i,t α i,t σ i,t K ( ν i, ) (cid:17) . y i,t − µ i,t ν i, (2 α i,t σ i,t K ( ν i, )) y i,t ≤ µ i,t )+ ν i, + 11 + ν i, (cid:16) y i,t − µ i,t − α i,t ) σ i,t K ( ν i, ) (cid:17) . y i,t − µ i,t ν i, (2(1 − α i,t ) σ i,t K ( ν i, )) y i,t > µ i,t ) . (25)The elements of the information matrix corresponding to location parametersare given by I µi,t = E (cid:2) ∆ µi,t ∆ µ (cid:48) i,t | Y t − (cid:3) = 1 σ i,t (cid:104) ν i, + 1 α i,t ( ν i, + 3) K ( ν i, ) + ν i, + 1(1 − α i,t )( ν i, + 3) K ( ν ,i ) (cid:105) . (26)The score with respect to scale parameters is∆ σi,t = ∂ log p ( y i,t | Y t − ) ∂σ i,t = (cid:2) ( ν i, + 1)1 + ( y i,t − µ i,t α i,t K ( ν i, ) σ i,t √ ν i, ) × ( y i,t − µ i,t α i,t σ i,t K ( ν i, ) √ ν i, ) − (cid:3) /σ i,t y i,t < µ i,t ) , + (cid:2) ( ν i, + 1) 11 + ( y i,t − µ i,t − α i,t ) K ( ν i, ) σ i,t √ ν i, ) × y i,t − µ i,t − α i,t ) σ i,t K ( ν i, ) √ ν i, ) − (cid:3) /σ i,t y i,t > µ i,t ) . (27)The elements of the information matrix corresponding to scale parameters are givenby I σi,t = E (cid:2) ∆ σi,t ∆ σ (cid:48) i,t | Y t − (cid:3) = 2 σ i,t (cid:104) α i,t ν i, ν i, + 3 + (1 − α i,t ) ν i, ν i, + 3 (cid:105) . (28)The score with respect to shape parameters is∆ αi,t = ∂ log p ( y i,t | Y t − ) ∂α i,t = ν i, + 1 ν i,
11 + ( y i,t − µ i,t α i,t K ( ν i, ) σ i,t √ ν i, ) × (( y i,t − µ i,t σ i,t K ( ν i, ) √ ν i, ) α i,t y i,t < µ i,t ) , + ν i, + 1 ν i,
11 + ( y i,t − µ i,t − α i,t ) K ( ν i, ) σ i,t √ ν i, ) × (( y i,t − µ i,t σ i,t K ( ν i, ) √ ν i, ) − α i,t ) y i,t > µ i,t ) . (29)The elements of the information matrix corresponding to shape parameters are givenby I αi,t = E (cid:2) ∆ αi,t ∆ α (cid:48) i,t | Y t − (cid:3) = 3 (cid:104) ν i, + 1 α i,t ( ν i, + 3) + ν i, + 1(1 − α i,t )( ν i, + 3) (cid:105) . (30)28he formulae for the information matrix can be found in Zhu and Galbraith (2010).Finally, by defining the vector a i,t = ( µ i,t , σ i,t , α i,t ) (cid:48) , the score with respect to thetime-varying parameters of series i is:∆ i,t = ∂ log p ( y i,t | Y t − ) ∂f t = ∂ log p ( y i,t | Y t − ) ∂a i,t . ∂a i,t ∂f t (31)while the information matrix isE (cid:2) ∆ i,t ∆ (cid:48) i,t | Y t − (cid:3) = (cid:16) ∂a i,t ∂f t (cid:17) (cid:48) I µi,t I µ,αi,t I σi,t I µ,αi,t I αi,t ∂a i,t ∂f t . (32) Appendix B Summary of the model specifications a b l e : D e s c r i p t i o n o f t h e m o d e l s M o d e l L a b e l C o nd i t i o n a l d i s tr i bu t i o n s Sp ec i fi c a t i o n s f o r l o c a t i o n , s c a l e a nd s h a p e p a r a m e t e r s D a t a ( a ) A S T D F M w i t hS c a l e + Sh a p e C F s A s y mm e tr i c S t ud e n t - t( A S T ) f o r G D P ; S k e w e dS t ud e n t - t f o r I P , a nd I W H ; N o r m a l f o r W E I . D y n a m i c f a c t o r m o d e l s f o r l o c a t i o n , s c a l e a nd s h a p e p a r a m e t e r s . Q u a rt e r l y G D P ; r o lli n g q u a rt e r l y I P a nd I W H ; m o n t h l y W E I ( b ) A S T D F M ( S k C o v s ) w i t hS c a l e + Sh a p e C F s A s y mm e tr i c S t ud e n t - t( A S T ) f o r G D P ; S k e w e d N o r m a l f o r I P , a nd I W H ; N o r m a l f o r W E I . D y n a m i c f a c t o r m o d e l s f o r l o c a t i o n , s c a l e a nd s h a p e p a r a m e t e r s . Q u a rt e r l y G D P ; r o lli n g q u a rt e r l y I P a nd I W H ; m o n t h l y W E I ( c ) D F M w i t hS V a ndS c a l e C F N o r m a l f o r a ll s e r i e s D y n a m i c f a c t o r m o d e l f o r l o c a t i o n s ( m e a n s ) a nd s c a l e s ( v o l a t ili t i e s ) ; Q u a rt e r l y G D P ; m o n t h l y I P , I W H a nd W E I ( d ) A S T D F M ( S k C o v s ) w i t hS V a nd T V s h a p e A s y mm e tr i c S t ud e n t - t( A S T ) f o r G D P ; S k e w e d N o r m a l f o r I P , a nd I W H ; N o r m a l f o r W E I . D y n a m i c f a c t o r m o d e l f o r l o c a t i o n s ; R a nd o m w a l k m o d e l s f o r s c a l e a nd s h a p e p a r a m e t e r s . Q u a rt e r l y G D P ; m o n t h l y I P , I W H a nd W E I ( e ) D F M w i t hS VN o r m a l f o r a ll s e r i e s D y n a m i c f a c t o r m o d e l f o r l o c a t i o n s ( m e a n s ) a nd s t o c h a s t i c s c a l e s ( v o l a t ili t i e s ) . Q u a rt e r l y G D P ; m o n t h l y I P , I W H a nd W E I ( f ) S t - t D F M w i t hS V S t ud e n t - t f o r G D P , I P , a nd I W H ; N o r m a l f o r W E I . D y n a m i c f a c t o r m o d e l f o r l o c a t i o n s ( m e a n s ) a nd s t o c h a s t i c s c a l e s ( v o l a t ili t i e s ) . Q u a rt e r l y G D P ; m o n t h l y I P , I W H a nd W E I N o t e : T h e s k e w e dn o r m a l d i s tr i bu t i o nh e r e i s d e r i v e db y c o n s tr a i n i n g t h e A S T d i s tr i bu t i o n t o h a v e G a u ss i a n t a il p a r a m e t e r s ( ν = ν = ∞ ) . T h e s k e w e dS t ud e n t - t i s d e r i v e db y c o n s tr a i n i n g t h e A S T d i s tr i bu t i o n t o h a v e a un i q u e t a il p a r a m e t e r( ν = ν ) . T h e S t ud e n t - t i s d e r i v e db y c o n s tr a i n i n g t h e A S T d i s tr i bu t i o n t o h a v e G a u ss i a n s h a p e a nd t a il p a r a m e t e r s ( α = . nd ν = ν = ∞ ) . I P = I ndu s tr i a l P r o du c t i o n ; I W H = I nd e x o f w o r k i n g h o u r s ; W E I = W ee k l y E c o n o m i c I nd e xx