Gil Dafnai , Jonathan Sidi
Research Department, Bank of Israel
Motivation : GDP data is being published at a six week lag after the end of the relevant quarter (and it is needed sooner).
However : There is a lot of monthly data that is available before the policy meetings.
Therefore : We use real-time monthly data in order to Nowcast the GDP 3 weeks ahead of publication.
General Data Set ( ): ◦ 170 monthly Indicators: 95% Domestic and 5%
Global. History:
◦ All series begin at least at 1998Q1. Endpoints:
◦ All series have value for at least two month of the projected quarter.
1. Seasonal adjustment by X12-ARIMA 2. Holt and Winters exponential smoother is
applied where necessary3. Convert to lower frequency (quarterly) by
average observation 4. Convert to percent change 5. Standardize
The resulting sample size is defined as ̂
ˆ d
Conditional Selection Methods
Unconditional Selection Methods
Multiple Univariate
LASSO Elastic Net
SparsePCA
PCA
1. Two Component Norm2. Iterated Component3. Selected Loadings Stepwise
Regression
Stepwise Regression
dIntermediary
StepFinalStep
Benchmark Method
Conditional Selection Methods
Unconditional Selection Methods
Multiple Univariate
LASSO Elastic Net
Stepwise Regression
Stepwise Regression
Intermediary Step
FinalStep
ˆ d
d
1. RUN:2. Calculate AICi - and keep the top 25 (in Z)
3. Run Stepwise Backward regression on:
4. Calculate Static Forecast
,ˆ ˆ;d i i d iGDP c
25
1
ˆd i i
i
GDP c Z
PCA and SPCA
Conditional Selection Methods
Unconditional Selection Methods
Multiple Univariate
LASSO Elastic Net
SparsePCA
PCA
1. Two Component Norm2. Iterated Component3. Selected Loadings Stepwise
Regression
Stepwise Regression
Intermediary Step
FinalStep
ˆ d
d
Original Data Centered Data
Rotated Centered Data Projection on max variance axis
SC1
Too many variables causes the inference to be extremely difficult
What characteristic do PC1 or PC2 represent in the data???
Same Data Set!!!
Retail Sales Indices
1. IL and US Consumer Confidence
2. Purchasing Manager’s Indices
The amount of variance in the data that is explained by the PCs decreases as sparsity increases
22
, , 1, ,ˆ1 1 1
ˆ argmini
p pn
i spca i i i j j i ji j j
SC X
L2-norm L1-norm
2 2`, 1, ,ˆ
1 1 1
ˆˆ , argmini
p pn
i i i j j i ji j j
X X
Is the ith row of the data matrix XiX
Three Methods1. The classic approach
(dimension reduction)2. Two component norm (TCN)
(variable selection)
3. Iterated component (IC) (Jolliffe 1973)
(variable selection)
Loadings Matrix
1,1 1,2 1,
2,1 2,2 2,
,1 ,2 ,
p pL
p
p
p p p p
l l l
l l l
l l l
LASSO and Elastic Net
Conditional Selection Methods
Unconditional Selection Methods
Multiple Univariate
LASSO Elastic Net
Stepwise Regression
Stepwise Regression
Intermediary Step
FinalStep
ˆ d
d
1
ˆ. :p
jj
s t t
21
ˆ. :p
jj
s t t
2ˆ1
ˆ ˆargminj
n
i
Y X
1. Advantages◦ General form of algorithm makes it applicable to
many problems in econometrics.◦ Ability to produce decomposition of variable
contribution of the forecast.
2. Shortcomings◦ Can not select more then n variables◦ If n>p then ridge is better◦ No grouping
LASSO Conclusions
Conditional Selection Methods
Unconditional Selection Methods
Multiple Univariate
LASSO Elastic Net
Stepwise Regression
Stepwise Regression
Intermediary Step
FinalStep
ˆ d
d
2 2
ˆ1 1
ˆ argmin 1j
pn
EN j ji j
Y X
RIDGE LASSO
L2-norm L1-norm
25% 50% 75%
q75 q50
Bias
CBS
2 2
ˆ1 1
ˆ argmin 1j
pn
EN j ji j
Y X
2 2
ˆ1 1
ˆ argmin 1j
pn
EN j ji j
Y X
Seasonally Adjusted, Annual Percent Change
First Release will be used as control group
Comparison of Variable Selection Consistency (24 Periods)
Tomer Kriaf
Research Department, Bank of Israel
Consumption Equation:Import of Durables, VAT, Confidence Index, Revenue Index (L), Imports of Raw Materials, TA Stock Market Index.
Fixed Capital Formation Equation:Imports of investment Goods, Capital Utilization, PMI, lagged Inventories, TA Stock Market Index.
Inventories Equation:Exports of goods, Revenue Index, Industrial Production Index.
Exports Equation:Exports of Goods, PMI-USA.
Import Equation:Imports of Goods, Imports of Services.
GDP Equation:Derived GDP, Indirect Tax, Income Tax, TA Stock Market Index.
De
riv
ed
GD
P
-6.0
-4.0
-2.0
0.0
2.0
4.0
6.0
8.0
10.0
2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2
2004 2005 2006 2007 2008 2009 2010
תחזית מוץ למדגם התוצר הנאמד בתוך המדגם In Sampleתמ"ג בפועל Out of Sample Actual
-5.0
-4.0
-3.0
-2.0
-1.0
0.0
1.0
2.0
3.0
4.0
5.0
2009Q2 2009Q3 2009Q4 2010Q1 2010Q2
תחזית נאיבית NOWCAST בזמן אמת I אומדן למ"ס אומדן למ"ס אחרוןPath Forecast
Real Time Nowcast CBS Current Release
CBS First Release
Gil Dafnai and Jonathan Sidi
Research Department
Bank of Israel