High Dimensional Data

Outline

• Dimensionality Reduction
• Factor Models
• Dynamic Factor Models
• Nowcasting

High Dimensional Data

• For most economic and many business forecasting tasks, a lot of potentially relevant information is available
  • FRED has over 500,000 economic time series
  • An individual business may have hundreds or thousands of product lines, establishments, workers, etc
• Majority of techniques we have looked at either only use the series whose forecast we are targeting
  • Or add a small set of carefully chosen predictors, chosen based on knowledge of economic or business process
• Can we do better by using more of the data available to us?
  • It seems like the extra information should help: even if only slightly related, forecast should improve
• But handling many predictors also creates big problems for existing methods
  • Adding predictors adds noise, increasing overfitting
  • If many or most predictors don’t help much, adding them could just make predictions worse
  • With as many series as values of outcome, can always get a perfect fit even if all predictors pure nonsense
• Can we find a way to get benefits from the extra information in a reliable way?
  • Today: Some classic approaches
  • Next two classes: Some more modern methods

Setup

• Goal is to forecast \(y_{T+1}\) using available data, including past values \(\mathcal{Y}_{T}=\{y_t\}_{t=1}^{T}\)
• Also have access to \(N\) additional variables each period, \(X_t=(x_{1,t},x_{2,t},\ldots,x_{N,t})^{\prime}\in\mathbb{R}^{N}\)
  • \(X=(X_1^{\prime},X_2^{\prime},\ldots,X_T^{\prime})\in\mathbb{R}^{T\times N}\) collects all the data into an array
• \(N\) may be large: comparable to or bigger than \(T\)
  • Any approach which uses all series in \(X_t\) without restrictions to predict \(y_{t+h}\) likely to overfit badly
  • This is without even getting into also using past values of \(X_t\)
• Goal is to come up with forecast procedures \(\widehat{f}\) and classes of forecasting rules \(\mathcal{F}(X)\) which ensure good performance
  • Balance Risk \(R^{*}_p(\mathcal{F})=\underset{f\in\mathcal{F}(X)}{\inf}E_p\ell(y_{T+h},f(X_{T}))\) and overfitting \(\frac{1}{T-h}\sum_{t=1}^{T-h}\ell(y_{t+h},\widehat{f}(X_t))-R^{*}_p(\mathcal{F})\)
  • Alternately, construct probability model over \((y_t,X_t)\) which provides accurate description of entire system

Dimensionality Reduction

• The most classical approach to the problem of working with high-dimensional data is dimensionality reduction
• Idea: although \(N\) large, most information in \(X\) can be compressed into a summary which has a much smaller dimension
• Instead of working with \(X_t\in\mathbb{R}^{N}\) work with \(F_t=(f_{1,t},f_{2,t},\ldots,f_{d,t})\in\mathbb{R}^d\)
  • \(d\) can be much smaller than \(N\)
• \(F_t\) is calculated from \(X\), and replaces it in forecasting procedure
  • Make forecasts using \((\mathcal{Y}_T,F)\), rather than \((\mathcal{Y}_T,X)\)
• Choose \(F_t\) is a reasonable number of dimensions and contains the relevant info in \(X\)
  • Preserve information gain from using large set of series
  • Avoid complexity and overfitting from overly complicated models

Dimensionality Reduction Approaches

• 2 ways of finding a summary \(F_t\in\mathbb{R}^d\)
• Unsupervised Dimensionality Reduction: \(F_t\) calculated only using information in \(X\)
  • Reflects some intrinsic structure in the predictors, which are related in a way which makes info beyond summary redundant
• Example: Sample average: \(F_t=\frac{1}{N}\sum_{i=1}^{N}x_{i,t}\)
  • Use if all series are basically redundant copies of the same information, with differences not relevant
  • E.g., instead of including 6 different measures of money supply, take average of all of them
• Supervised Dimensionality Reduction: Use information in \(\mathcal{Y}_T\) and \(X\) to determine \(F_t\)
  • \(X\) may have a lot going on, but only a bit of that info is related to \(y\)
• Example: Lasso. Regress \(y_{t}\) on \(X_{t}\) with L1 penalty on coefficients
  • Sets to 0 many or most coefficients: predictor is combination of only a small subset of columns of \(X\)
• Both approaches popular and effective in economics
  • Unsupervised case useful when intrinsic dimension of \(X\) is low: knowing only \(F_t\), one could get an accurate guess of \(X_t\)
    • No point in including full set of predictors since you could recover it from smaller set anyway
  • Supervised case useful when \(X\) might not have simple structure, but parts that are related to \(y\) do

Factor Models: Static Case

• Unsupervied approach to get \(d\) factors \(F_t=\{f_{j,t}\}_{j=1}^{d}\), which predict \(\{X_{i,t}\}_{i=1}^{N}\) where \(N>>d\)
• Want to minimize squared error loss of predictions using some linear combination of \(d\) factors
• Find coefficients \(\Lambda\), called loadings and factors \(F\) which give best prediction
  • \(V(\Lambda,F)=\frac{1}{NT}\sum_{i=1}^{N}\sum_{t=1}^{T}(x_{i,t}-\sum_{j=1}^{d}\lambda_{i,j}f_{j,t})^2\)
  • Equivalent to log likelihood in model \(X_t=\Lambda F_t+e_t\), \(\{e_{i,t}\}_{i=1}^{N}\overset{iid}{\sim}N(0,\sigma_e^2)\)
• Trick is that although neither \(\Lambda\) nor \(F\) is known, we can estimate both together by minimizing \(V(\Lambda,F)\) jointly
  • \((\widehat{\Lambda},\widehat{F})\in\underset{\Lambda\in\mathbb{R}^{N\times d},F\in\mathbb{R}^{d\times T}}{\arg\min}V(\Lambda,F)\)
• Combination of factors and loadings in \(d\) dimensions minimizing MSE is not uniquely determined
  • Many possible \(F\) and \(\Lambda\) could all give exactly the same predictions
  • E.g., divide \(f_3\) by 2 and multiply \(\lambda_3\) by \(2\) and predictions are the same
  • General Case: \((\Lambda R^{-1})(RF)\) for any \(d\times d\) rotation matrix R defines a valid factor decomposition
• For prediction purposes, we rarely care which factor is which, so long as we are predicting with all of them
  • Can choose an arbitrary minimizer, then rotate (pick \(R\)) to find any others if needed

Standard Estimator: Principal Components Analysis

• A convenient normalization is to assume loadings \(\lambda_j\) are uncorrelated with each other and have norm \(1\)
  • Typically order them from most to least variance explained
  • Factor estimates found in this way are called principal components of \(X\)
• Factor decomposition with this normalization is called Principal Components Analysis or PCA
  
• The principal components are linear combinations of the observations, chosen in such a way that the largest possible amount of (remaining) variation in the data is explained by each
• Let \(\widehat{\lambda}_1=\underset{\lambda\in\mathbb{R}^{N}:\left\Vert\lambda\right\Vert=1}{\arg\max}\lambda^{\prime}X^\prime X\lambda\): then \(\widehat{f}_1=X\widehat{\lambda}_1\) is the first principal component of data \(X\)
  • \(\widehat{f}_1\) is the norm 1 linear combination of \(x_{i}\)’s with maximum variance
• Each subsequent principal component is defined analogously, except maximizing the proportion explained of the residual variance after previous ones have been removed
  • If you have done linear algebra: \(\widehat{\lambda}\) are the eigenvectors of \(X^{\prime}X\), the covariance matrix of data
  • \(\widehat{Var}(\widehat{f}_{j})\), the variance which is maximized by the choice, is the corresponding eigenvalue
• Principal Components Analysis (PCA) solution mostly useful because easily computed, due to linear algebra connections
• If for interpretation purposes another normalization is preferred, can achieve it by rotation of the PCA solution

Choosing the number of components

• To see how number of factors affects prediction, use graphical measure called a scree plot
• It plots the number of factors against the maximum fraction of variance explained by adding one more factor
  • Technical definition for linear algebra fans only: it plots the eigenvalues of the covariance matrix, in order
• If model is well described by a few factors, scree plot will flatten off after those factors
• Can also treat problem like model selection, and choose a number using an information criterion
• Bai and Ng (2002) suggested \(IC^{\text{Bai-Ng}}(d)=IC^{\text{Bai-Ng}}(d)=\ln (V(\widehat{\lambda},\widehat{F})) +dg(N,T)\)
  • \(g(N,T)\) is some function of N and T, e.g. \(\frac{N+T}{NT}\ln(\frac{NT}{N+T})\)
• Choosing d to minimize \(IC^{\text{Bai-Ng}}(d)\) selects a set of factors which provide good performance without overfitting

library(dplyr) #Data Manipulation
library(ggplot2)

#Load Current FRED-MD Series, as downloaded from https://research.stlouisfed.org/econ/mccracken/fred-databases/
# Described in McCracken and Ng (2014) http://www.columbia.edu/~sn2294/papers/freddata.pdf

#Series of 729 observations of 129 macroeconomic indicators
currentMD<-read.csv("Data/current.csv")
#Drop data column, so result contains only data
fmd<-select(currentMD,-one_of("sasdate"))
#Turn into time series
fredmdts<-ts(fmd,frequency=12,start=c(1959,1))

#Take PCA of Scaled Series
mdPCA<-prcomp(~.,data=fmd,na.action=na.exclude,scale=TRUE)

#Compute statistics
mdPCAinfo<-summary(mdPCA)

#Collect all Group 6 Variables: These are financial series
finseries<-data.frame(currentMD$FEDFUNDS,currentMD$CP3Mx,currentMD$TB3MS,currentMD$TB6MS, currentMD$GS1,currentMD$GS5,currentMD$GS10, currentMD$AAA,currentMD$BAA,currentMD$COMPAPFFx,currentMD$TB3SMFFM,currentMD$TB6SMFFM,
                      currentMD$T1YFFM,currentMD$T5YFFM,currentMD$T10YFFM,currentMD$AAAFFM,
                      currentMD$BAAFFM,currentMD$TWEXMMTH,currentMD$EXSZUSx,
                      currentMD$EXJPUSx,currentMD$EXUSUKx,currentMD$EXCAUSx)

finpca<-prcomp(~.,data=finseries,na.action=na.exclude,scale=TRUE)

#Alternate PCA Approaches in R

?prcomp #Default PCA in stats library
?princomp #As above, but with slightly different scaling?

?screeplot #Use to investigate magnitude of components

# Syntax example:

pc.cr <- princomp(USArrests, cor = TRUE)
screeplot(pc.cr)

#Nonlinear dimension reduction tools
library(Rtsne) #Library for t-SNE: 2d representation, for visualization
library(diffusionMap) #Diffusion Maps: nonlinear low dimensional representation from distance matrices (rather than covariance matrices between points) (cf adapreg for) regression on diffusion map components
library(kernlab) #Contains kernel PCA (and spectral clustering)

#To investigate
#install.packages("bayesdfa") #Bayesian dynamic factor models! Great, but want frequentist ones too
#install.packages("nowcasting") #Dynamic factor models for nowcasting!
#install.packages("tsfa") #Time Series factor models

Example: Principal Components Decomposition

• Use FRED-MD (McCracken & Ng 2014) Database of macro indicators available on FRED
  • 129 Macroeconomic series, 728 observations
• Take PCA of (normalized to mean 0 variance 1) series using prcomp command (princomp also can be used)
• First principal component accounts for 52.552 percent of variability in observations
• First 2 account for 71.938, 3 for 80.721, etc
• Over 99% Accounted for in first 20: can go from 129 data series to 20 dimensions, and lose very little variation in data
  • The vector of principal components can replace the full 129-dimensional observation \(x_t\)
• Factor loadings or rotations are vector of weights placed on each of the 129 series to produce each principal component

2 Dimensional PCA Approximation of Data Set

#You can use default biplot(mdPCA) to get a version of this, including arrows displaying loadings, but it is too busy to read
# cf https://stackoverflow.com/questions/6578355/plotting-pca-biplot-with-ggplot2

data <- data.frame(obsnames=row.names(mdPCA$x), mdPCA$x)
plot <- ggplot(data, aes_string(x="PC1", y="PC2")) + geom_text(alpha=.4, size=3, aes(label=obsnames))+
        ggtitle("First 2 Principal Components of Macro Series",subtitle="Data Labels are Time Period")
plot

Alternatives: Nonlinear Methods and Sparsity

• Most common dimensionality reduction approaches and corresponding predictions are linear
  • \(X_t\) defined in terms of linear function of unknown components \(F_t\)
• It is completely possible to allow a nonlinear representation instead
  • Can have \(X_t\approx T(F_t,\theta)\) for some nonlinear \(T()\), with parameters learned from data
• Finding a good nonlinear transformation can be difficult, as not all cases preserve structure of data
• Case allowing for general nonlinearities called “manifold learning”
  • Variety of approaches exist: diffusion maps (diffusionMap in R), isomap, LLE, kernel PCA, etc
  • Autoeencoders use neural network functions as map: recently most popular method
    
• All used in similar way to PCA, to extract low-dimensional set of features to use for visualization and prediction
• Alternative approach is to just pick small subset of series: sparse representation
  • Problem is that you may not know which series best describe \(X_t\) and/or \(y_t\)
    • More useful if sparsity pattern learned in supervised approach
  • If all series highly correlated, as in macroeconomics, sparse representation may be worse than factor representation at same \(d\)

Factor Models: Dynamic Case

• Factor model approach useful for getting current summary of \(X_t\)
• To leverage this for forecasting, use Dynamic Factor Model, which uses past factors to predict future ones
• X still depends on factors, and factors \(F_t\) follow vector autoregression
  • \(X_t=\Lambda F_t+e_t\), \(F_{t}=AF_{t-1}+u_t\)
• Format is exactly that of of a state space model, with \(d\)-dimensional state, \(N\) dimensional observations
• If we assume \(e_{i,t}\overset{iid}{\sim} N(0,\sigma^2_e)\), \(u_{j,t}\overset{iid}{\sim} N(0,\sigma^2_u)\), can estimate and predict using Kalman Filter
  • Estimate like any other state space model, and predict using filtered estimates \(\widehat{F}_{T}\) of factors
  • Use likelihood for Bayesian version with priors on coefficients, cf R library bayesdfa
• Can allow additional lags of factors in model, or including standard state space components like trends, seasons, etc
• Would it help to predict \(X_t\) using lags of \(F_{t-j}\), i.e. \(X_t=B(L) F_t+e_t\) for some lag polynomial?
  • It turns out, not needed: if lags matter for \(X_t\), they show up as extra factors
  • But, in this case, dimension of shocks \(u_t\) may be smaller than dimension \(d\) of factors, so allow for this in model

Two Step Estimation of Dynamic Factor Model

• An easier and more robust method exploits fact that \(X\) still has factor structure
  • Step 1: Use PCA on \(X\) to obtain \(\{\widehat{F}_t\}_{t=1}^{T}\) in \(X_t=\Lambda F_t+e_t\)
  • Step 2: Estimate VAR in \(\widehat{F}_T\) to obtain \(A\) in \(\widehat{F}_{t}=A\widehat{F}_{t-1}+u_t\)
• Usefulness of this method is that PCA gives good predictions even if \(X_t\) doesn’t have exact factor structure
  • \(e_t\) can have some remaining correlations between series even after \(\Lambda F_t\) taken out
  • Can also have \(e_{i,t}\) autocorrelated, reflecting predictability not captured by \(F_t\)
• Extra error introduced due to fact that factors estimated, but if \(N,T\) both large, this is small
• Do need stationarity, weak dependence for good estimates, as usual for ERM
  • Since needed for all series in \(X_t\), be careful to difference or detrend each before estimation
• In practice, extra robustness often results in better performance than Bayesian estimate based on exact factor structure
• Bayesian approach useful if \(T\) or \(N\) not super big, so factors estimated imprecisely
  • Also can handle nonstationary factors, if form of nonstationarity specified correctly

Factor-Augmented Regression

• To predict series \(y_{t+h}\) which is not in \(X\) using dynamic factors, simply include \(\widehat{F}_t\) as predictors
  • Estimate \(\widehat{F}_t\) by PCA, which is appropriate in static or dynamic case
• Use predictor class \(\mathcal{F}=\{f(\widehat{F},\beta)=\beta_0+\sum_{j=1}^{d}f_j\beta_{j}:\ \beta\in\mathbb{R}^{d+1}\}\) inside ERM or other estimator
• Like standard ERM with minor caveat that predictors no longer given but first estimated using the data
• Properties are similar, in terms of risk: under stationarity, weak dependence, etc, risk converges to oracle risk
• Can also add \(\widehat{F}_t\) along with a set of other, carefully chosen predictors in a Factor Augmented VAR (FAVAR)
  • Allows using additional idiosyncratic info in small set of series, along with correlated info in others, to predict outcome of interest
• Extremely popular in policy circles
  • Jim Stock, one of developers of Dynamic Factor Model, was head of Council of Economic Advisors
  • FAVAR developed by group including former Fed Chair Ben Bernanke
    • Used to analyze interaction of monetary policy, inflation, and employment, which Fed is mandated to take into account
    • Allows incorporating huge amount of other series used by Fed

## Code to Fit Bayesian factor models
# Run this on your own time, as it is too slow to run each time I compile the file

#library(bayesdfa) #Bayesian Factor Models

# library(fredr) #Data Source
# fredr_set_key("8782f247febb41f291821950cf9118b6") #Key I obtained for this class
# 
# serieslist<-c("GS1M","GS3M","GS6M","GS1","GS2","GS3","GS5","GS7","GS10","GS20","GS30")
# 
# treasuryyields<-list()
# for (i in 1:length(serieslist)){
#   
#   val<-fredr(series=serieslist[i])
#   
#   treasuryyields[[i]]<-ts(val$value,frequency=12,names=serieslist[i],names=serieslist[i])
# }

# options(mc.cores = parallel::detectCores()) #Use parallel computing when running MCMC

### Warning: With Data Size this large, following command does not appear feasible on a laptop
## Command works fune on small data sets, but not on this one
## #Fit Bayesian Dynamic Factor Model using "bayesdfa"
## dynfactormodel<- fit_dfa(
##   y = t(finseries), num_trends = 3, zscore = TRUE,
##   iter = 2000, chains = 4, thin = 1
## )

# ## Alternate approach: try 1 factor model of just small set of interest series 
# # This is also very slow, but at least feasible. Total estimation takes 76 minutes on a 4-core laptop.
# fins1<-data.frame(currentMD$FEDFUNDS, currentMD$GS1,currentMD$GS5,currentMD$GS10)
# # Bayesian 1 factor model, 4 normalized series, with random walk assumed for factor dynamics 
# dynfactormodel<- fit_dfa(
#   y = t(fins1), num_trends = 1, zscore = TRUE,
#   iter = 2000, chains = 4, thin = 1
# )
# #Select a rotation of factors (irrelevant with 1, but this also computes summary stats)
# rt<-rotate_trends(dynfactormodel) 
# #Plot factor over time
# plot_trends(rt)
# #Plot predictions for each series
# plot_fitted(dynfactormodel)
# #Plot posterior estimates of factor loadings
# plot_loadings(rt)

Nowcasting

• A common problem when using many data series is that not all are available at the same time
• Some may be at different frequencies: e.g., quarterly, annual, monthly
  • Even within a month, some may come out earlier or later in month,
• Result is that current period information is incomplete, containing some series but not others
• Goal of Nowcasting is to use this incomplete info to predict a missing or external series
  • E.g., use monthly releases to predict quarterly GDP
• Good nowcasts can be achieved using slight variant of dynamic factor model
  • \(x_{i,t}=\sum_{j=1}^d \lambda_{i,j} f_{j,t}+u_t\) if \(i\) observed in period \(t\), otherwise it disappears
  • Factors still follow VAR \(F_t=AF_{t-1}+e_{t}\)
• Kalman filter allows inferring \(\widehat{F}_{t}\) even with missing data
  • Get good estimate of current factors with whatever data is available
  • Can “fill in” what a series would have been using current guess of factors plus known factor loadings

Nowcasting: One Approach

• “2 step procedure” due to Giannone, Reichlin, Small (2008), originators of procedure

1. Estimate parameters of dynamic factor model
   • 1. Estimate \(d\) factors \(F_t\) from series by PCA on \(X_t\)
   • 2. Aggregate monthly series up to quarterly data, and estimate loadings of \(y_{t}\) on factors by Least Squares
     • \(\widehat{\beta}=\underset{\beta}{\arg\min}\sum_{k=1}^{T/3}(y_{k}-\sum_{j=1}^{d}\beta_j\widehat{f}_{j,3k})^2\)
   • 3. Estimate VAR in \(F_t\), \(F_t=AF_{t-1}+e_{t}\) by OLS using PCA factors
2. Predict current factors using Kalman filter and use in forecast
   • 1. Run Kalman filter from dynamic factor model to get predicted factors \(\widehat{F}_{t}\) given current observations
   • 2. Predict \(\widehat{y}_{k}=\widehat{F}_{t}\widehat{\beta}\)

• Apply to set of monthly US economic indicators (1982-2007) from Giannone et. al. for predicting quarterly GDP
  • Implemented in nowcasting library
  • Select \(d\) by Bai-Ng Criterion, estimate model by above

library(nowcasting)
#Example syntax for nowasting based on quarterly factor model as in Giannone, Reichlin, Small (2011)


# Old Syntax: manually aggregate 
# #Use Stock-Watson US GDP and indicators data
# gdp <- month2qtr(x = USGDP$base[,"RGDPGR"])
# # Extract Real GDP Growth from data
# gdp_position <- which(colnames(USGDP$base) == "RGDPGR")
# #Aggregate monthly up to quarterly data
# capture.output(base <- Bpanel(base = USGDP$base[,-gdp_position],trans = USGDP$legend$Transformation[-gdp_position],aggregate = TRUE),file="/dev/null")

#New syntax (automatically aggregates)
data(USGDP)
gdp_position <- which(colnames(USGDP$base) == "RGDPGR")
base <- Bpanel(base = USGDP$base[,-gdp_position],
               trans = USGDP$legend$Transformation[-gdp_position],
               aggregate = TRUE)

## [1] "AVGWKLYCL" "CPOUT"     "M3"

usdata <- cbind(USGDP$base[,"RGDPGR"], base)
colnames(usdata) <- c("RGDPGR", colnames(base))
frequency <- c(4, rep(12, ncol(usdata) -1))

Diagnostics: Find order of factor model by Bai-Ng Criterion

#Diagnostics 1: Select number of Factors by Bai-Ng Information Criterion
factorselect<-ICfactors(base)

#Diagnostics 2: Select number of lags and shocks by another info criterion
shockselect<-ICshocks(base,r=factorselect$r_star)

Results: Nowcast

#Run 2 step nowcasting, with optimal # of factors, lags, and error terms

#Old Syntax
#now2sq <- nowcast(y = gdp, x = base, r = factorselect$r_star, p = shockselect$p, q = shockselect$q_star, method = '2sq')

#New Syntax
now2s <- nowcast(formula = RGDPGR ~ ., data = usdata, r = factorselect$r_star, p = shockselect$p, q = shockselect$q_star,
          method = '2s', frequency = frequency)

#Plot results
nowcast.plot(now2s,type="fcst")

Results: Factor Estimates

#Plot results
nowcast.plot(now2s,type="factors")

#Additional Displays, Not Shown

##Fraction of variance Explained by each factor
#nowcast.plot(now2s,type="eigenvalues")
## Display factor loadings of each series on Factor 1
#nowcast.plot(now2s,type="eigenvectors")

Application: Financial Times Nowcast

• Gavyn Davies at Financial Times reports nowcasts and forecasts global economic growth each month
  • Uses Bayesian Dynamic Factor Model due to Juan Antolin-Diaz et al at Fulcrum Asset Management
• Uses releases from all major economies to predict global and region-specific factors
• Bayesian state space model with unit roots, seasonality, heavy tailed errors and time-varying variance
  • Additional features improve uncertainty forecasts, account for correlation properties of data

Conclusions

• Many economic forecasting problems have more predictors available than can be incorporated in existing methods
• Simple solution is to reduce dimensionality of data by producing small set of combinations that contain most info
• Factor models use linear combinations, and can be estimated by Principal Components Analysis
  • Don’t take separate factors themselves as anything meaningful: only their combinations are determined by data
• Forecast using this information by adding factors as predictors or treating as a latent state
  • Dynamic Factor Model is special case of a linear state space model with fewer states than observables
• Factor models are currently dominant approach in applied macroeconomic forecasting
• Also useful for combining data at mixed frequency to “nowcast” series not yet updated

References

• Jushan Bai and Serena Ng (2002) “Determining the Number of Factors in Approximate Factor Models” Econometrica Volume 70, No.1, pp 191-221.
• Ben Bernanke, Jean Boivin, Piotr Eliasz (2005) “Measuring the Effects of Monetary Policy: A Factor-Augmented Vector Autoregressive (FAVAR) Approach” The Quarterly Journal of Economics, Volume 120, Issue 1, February, pp 387–422.
  • Introduced FAVARs
• Gavyn Davies “Global economy may still defy the pessimists this year” Financial Times, April 7, 2019
  • Access nowcasts without FT subscription via Fulcrum Asset Management
• Domenico Giannone, Lucrezia Reichlin, David Small (2008) “Nowcasting: The real-time informational content of macroeconomic data” Journal of Monetary Economics 55 pp 665-676.
• Michael McCracken and Serena Ng (2014) “FRED-MD: A Monthly Database for Macroeconomic Research”
• James Stock and Mark Watson (2002) “Forecasting Using Principal Components From a Large Number of Predictors” Journal of the American Statistical Association December, Vol 97, No 460
  • Describes factor-augmented regression forecasting