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ForecastingLecture 1: Foundations
Bruce E. Hansen
Central Bank of ChileOctober 29-31, 2013
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3-Day Course
TuesdayI Point ForecastingI Forecast SelectionI Leading IndicatorsI Variance ForecastingI Interval Forecasting
WednesdayI Combination ForecastsI Multi-Step ForecastingI Fan Charts
Thursday: Structural Breaks
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Course Website
www.ssc.wisc.edu/~bhansen/cbc
Slides for all lectures
Data for the lectures
R code for empirical analysis for lectures 1 & 2
Related: www.ssc.wisc.edu/~bhansen/creteI 5-day forecasting courseI All the empirical analysis reported here was done in spring 2012, so the“forecasts” appear to be about the past
I We can compare the forecasts with realizations
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Today’s Schedule
What is Forecasting?
Point Forecasting
Linear Forecasting Models
Forecast Selection: BIC, AIC, AICc , Mallows, Robust Mallows, FPE,Cross-Validation, PLS, LASSO
Leading Indicators
Variance Forecasting
Interval Forecasting
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Example 1
U.S. Quarterly Real GDPI 1960:1-2012:1
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Figure: U.S. Real Quarterly GDP
1960 1970 1980 1990 2000 2010
4000
6000
8000
1000
012
000
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Transformations
It is mathematically equivalent to forecast yn+h or any monotonictransformation of yn+h and lagged values.
I It is equivalent to forecast the level of GDP, its logarithm, orpercentage growth rate
I Given a forecast of one, we can construct the forecast of the other.
Statistically, it is best to forecast a transformation which is close to iidI For output and prices, this typically means forecasting growth ratesI For rates, typically means forecasting changes
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Annualized Growth Rate
yt = 400(log(Yt )− log(Yt−1))
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Figure: U.S. Real GDP Quarterly Growth
1960 1970 1980 1990 2000 2010
10
50
510
15
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Example 2
U.S. Monthly 10-Year Treasury Bill RateI 1960:1-2012:4
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Figure: U.S. 10-Year Treasury Rate
1960 1970 1980 1990 2000 2010
24
68
1012
14
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Monthly Change
yt = Yt − Yt−1
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Figure: U.S. 10-Year Treasury Rate Change
1960 1970 1980 1990 2000 2010
1.5
1.0
0.5
0.0
0.5
1.0
1.5
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Notation
yt : time series to forecastn : last observationn+ h : time period to forecasth : forecast horizon
I We often want to forecast at long, and multiple, horizonsI For the first days we focus on one-step (h = 1) forecasts, as they arethe simplest
In : Information available at time n to forecast yn+hI Univariate: In = (yn , yn−1, ...)I Multivariate: In = (xn , xn−1, ...) where xt includes yt , “leadingindicators”, covariates, dummy indicators
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Forecast Distribution
When we say we want to forecast yn+h given In,I We mean that yn+h is uncertain.I yn+h has a (conditional) distributionI yn+h | In ∼ F (yn+h |In)
A complete forecast of yn+h is the conditional distribution F (yn+h |In)or density f (yn+h |In)F (yn+h |In) contains all information about the unknown yn+hSince F (yn+h |In) is complicated (a distribution) we typically reportlow dimensional summaries, and these are typically called forecasts
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Standard Forecast Objects
Point Forecast
Variance Forecast
Interval Forecast
Density forecast
Fan Chart
All of these forecast objects are features of the conditional distribution
Today, we focus on point forecasts
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Point Forecastsfn+h|h, the most common forecast object“Best guess” for yn+h given the distribution F (yn+h |In)We can measure its accuracy by a loss function, typically squared error
` (f , y) = (y − f )2
The risk is the expected loss
En` (f , yn+h) = E((yn+h − f )2 |In
)The “best”point forecast is the one with the smallest risk
f = argminf
E((yn+h − f )2 |In
)= E (yn+h |In)
Thus the optimal point forecast is the true conditional expectationPoint forecasts are estimates of the conditional expectation
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Estimation
The conditional distribution F (yn+h |In) and ideal point forecastE (yn+h |In) are unknownThey need to be estimated from data and economic models
Estimation involvesI Approximating E (yn+h |In) with a parametric familyI Selecting a model within this parametric familyI Selecting a sample period (window width)I Estimating the parameters
The goal of the above steps is not to uncover the “true”E (yn+h |In),but to construct a good approximation.
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Information Set
What variables are in the information set In?All past lags
I In = (xn , xn−1, ...)
What is xt?I Own lags, “leading indicators”, covariates, dummy indicators
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Markov Approximation
E (yn+1|In) = E (yn+1|xn, xn−1, ...)I Depends on infinite past
We typically approximate the dependence on the infinite past with aMarkov (finite memory) approximation
For some p,
E (yn+1|xn, xn−1, ...) ≈ E (yn+1|xn, ..., xn−p)
This should not be interpreted as true, but rather as anapproximation.
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Linear Approximation
While the true E (yn+1|xn, ..., xn−p) is probably a nasty non-linearfunction, we typically approximate it by a linear function
E (yn+1|xn, ..., xn−p) ≈ β0 + β′1xn + · · ·+ β′pxn−p
= β′xn
Again, this should not be interpreted as true, but rather as anapproximation.
The error is defined as the difference between yn+h and the linearfunction
et+1 = yt+1 − β′xt
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Linear Forecasting Model
We now have the linear point forecasting model
yt+1 = β′xt + et+h
As this is an approximation, the coeffi cient and eror are defined byprojection
β =(E(xtx′t
))−1(E (xtyt+1))
et+1 = yt+1 − β′xtE (xtet+1) = 0
σ2 = E(e2t+1
)The conditional variance σ2t = E
(e2t+1|It
)may be time-varying
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Univariate (Autoregressive) Model
xt = (yt , yt−1, ..., yt−k+1)
A linear forecasting model is
yt+1 = β0 + β1yt + β2yt−1 + · · ·+ βkyt−k+1 + et+1
AR(k) —Autoregression of order kI Typical AR(k) models add a stronger assumption about the error et+1
F IID (independent)F MDS (unpredictable)F white noise (linearly unpredicatable/uncorrelated)
I These assumptions are convenient for analytic purpose (calculations,simulations)
I But they are unlikely to be true
F Making an assumption does not make the assumption trueF Do not confuse assumptions with truth
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Least Squares Estimation
β =
(n−1∑t=0
xtx′t
)−1 (n−1∑t=0
xtyt+1
)yn+1|n = fn+1|n = β
′xn
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GDP Example
yt = ∆ log(GDPt ), quarterlyAR(4) (reasonable benchmark for quarterly data)
yt+1 = β0 + β1yt + β2yt−1 + β3yt−2 + β4yt−3 + et+1
β s(β)Intercept 1.54 (0.45)∆ log(GDPt ) 0.29 (0.09)∆ log(GDPt−1) 0.18 (0.10)∆ log(GDPt−2) −0.05 (0.08)∆ log(GDPt−3) 0.06 (0.10)
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Point Forecast - GDP Growth
AR(4)
Data Forecast Actual2011:1 0.362011:2 1.332011:3 1.802011:4 2.912012:1 1.842012:2 2.59 1.20
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Interest Rate Exampleyt = ∆RatetAR(12) (reasonable benchmark for monthly data)
β s(β)Intercept −0.002 (0.01)∆Ratet 0.40 (0.06)∆Ratet−1 −0.26 (0.07)∆Ratet−2 0.11 (0.06)∆Ratet−3 −0.07 (0.07)∆Ratet−4 0.10 (0.07)∆Ratet−5 −0.08 (0.07)∆Ratet−6 −0.05 (0.06)∆Ratet−7 −0.09 (0.06)∆Ratet−8 −0.01 (0.07)∆Ratet−9 0.03 (0.07)∆Ratet−10 0.09 (0.07)∆Ratet−11 −0.08 (0.06)
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Point Forecast - 10-year Treasury Rate
AR(12)
Data Forecast ActualLevel Change Level Change Level Change
2012:1 1.97 -0.012012:2 1.97 0.002012:3 2.17 0.202012:4 2.05 -0.122012:5 1.93 -0.12 1.82 -0.23
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Forecast Selection
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Forecast SelectionWe used (arbitrarily) an AR(4) for GDP,and an AR(12) for the 10-year rateThe forecasts will be sensitive to this choiceGDP Example
Model ForecastAR(0) 2.99AR(1) 2.59AR(2) 2.65AR(3) 2.68AR(4) 2.59AR(5) 2.83AR(6) 2.83AR(7) 2.83AR(8) 2.78AR(9) 2.87AR(10) 2.87AR(11) 2.91AR(12) 3.45Bruce Hansen (University of Wisconsin) Foundations October 29-31, 2013 30 / 108
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Forecast Selection - Big Picture
What is the goal?I Accurate Forecasts
F Low Risk (low MSFE)
Finding the “true”model is irrelevantI The true model may be an AR(∞) or have a very large number ofnon-zero coeffi cients
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Testing
It is common to use statistical tests to select empirical models
This is inappropriateI Tests answer the scientific question: Is there suffi cient evidence toreject the hypothesis that this coeffi cient is zero?
I Tests are not designed to answer the question: Which estimate yieldsthe better forecast?
This is not a minor issueI Lengthy statistics literature documenting the poor properties of "postselection" estimators.
I Estimators based on testing have particularly bad properties
Tests are appropriate for answering scientific questions aboutparameters
Standard errors are appropriate for measuring estimation precision
For model selection, we want something different
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Model Selection: Framework
Set of estimates (models)
I β(m), m = 1, ...,M
Corresponding forecasts fn+1|n(m)
There is some population criterion C (m) which evaluates theaccuracy of fn+1|n(m)
I m0 = argminm C (m) is infeasible best estimator
There is a sample estimate C (m) of C (m)
m = argminm C (m) is empirical analog of m0β(m) is selected estimator
fn+1|n(m) selected forecast
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Point Forecast and MSFEGiven an estimate β(m) of β , the point forecast for yn+1 is
fn+1|n = β(m)′xn
The forecast error is
yn+1 − fn+1|n = x′nβ+ et+1 − x′n β(m)
= en+1 − x′n(
β(m)− β)
The mean-squared-forecast-error (MSFE) is
MSFE (m) = E(en+1 − x′n
(β(m)− β
))2' σ2 + E
((β(m)− β
)′Q(m)
(β(m)− β
))where Q(m) = E (xnx′n) .A good forecast has low MSFE
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Selection Criterion
Bayesian Information Criterion (BIC)I C (m) = P (m is true)
Akaike Information Criterion (AIC), Corrected AIC (AICc )I C (m) = KLIC
Mallows, Predictive Least Squares, Final Prediction Error,Leave-one-out Cross Validation:
I C (m) = MSFE
LASSOI Penalized LS
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Important: Sample must be constant when comparingmodels
This requires careful treatment of samples
Suppose you observe yt , t = 1, ..., n
Estimation of an AR(k) requires k initial conditions, so the effectivesample is for obserations t = 1+ k , ..., n
The sample varies with k, sample size is n− kFor valid comparison of AR(k) models for k = 1, ...,K
I Fix sample with observations t = 1+K , ..., nI n−K observationsI Estimate all AR(k) models using this same n−K observations
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Bayesian Information Criterion (BIC)
M models, equal prior probability that each is the “true”model
Compute posterior probability that model m is true, given data
Schwarz showed that in the normal linear regression model theposterior probability is proportional to
p(m) ∝ exp(−BIC (m)
2
)BIC (m) = n log σ2(m) + log(n)k(m)
whereI k(m) = # of parametersI σ2(m) = n−1 ∑n−1t=0 e
2t+1(m) = MLE estimate of σ2 in model m
The model with highest probability maximizes p(m),or equivalently minimizes BIC (m)
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Bayesian Information Criterion - Properties
ConsistentI If true model is finite dimensional, BIC will identify it asymptotically
ConservativeI Tends to pick small models
Ineffi cient in nonparametric settingsI If there is no true finite-dimensional model, BIC is sub-optimalI It does not select a finite-sample optimal model
We are not interested in “truth”, rather we want good performance
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Akaike Information Criterion (AIC)
Estimates Kullback-Leibler information criterion (KLIC) distancebetween true and estimated density
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AIC
In the linear regression model
AIC = 2L(θ) + 2k= n log σ2(m) + 2k(m)
Similar in form to BIC, but “2” replaces log(n)
Picking a model with the smallest AIC is picking the model with thesmallest estimated KLIC.
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Corrected AIC
In the normal linear regression model, Hurvich-Tsai (1989) calculatedthe exact AIC
AICc (m) = AIC (m) +2k(m) (k(m) + 1)n− k(m)− 1
Works better in finite samples than uncorrected AIC
It is an exact correction when the true model is a linear regression,not time series, with iid normal errors.
In time-series or non-normal errors, it is not an exact correction.
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Comments on AIC Selection
Widely used, partially because of its simplicity
Full justification requires correct specificationI normal linear regression
Critical specification assumption: homoskedasticityI AIC is a biased estimate of KLIC under heteroskedasticity
Criterion: KLICI Not a natural measure of forecast accuracy.
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Mallows Criterion
Cn(m) = σ2(m) +2n
σ2k(m)
Uses a preliminary estimate σ2 of the variance
Cn(m) is an (approximately) unbiased estimate of the MSFE underhomoskedasticity and one-step forecasting
Model m which minimizes Mallows criterion is an estimate of thelowest MSFE model
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Final Prediction Error (FPE) Criterion
FPEn(m) = σ2(m)(1+
2nk(m)
)
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Relation between Mallows, FPE, and Akaike
Take log of FPE and multiply by n
n log (FPEn(m)) = n log(
σ2(m))+ n log
(1+
2nk(m)
)' n log
(σ2(m)
)+ 2k(m)
= AIC (m)
Thus Mallows, FPE and Akaike model selection are quite similar
Mallows, FPE, and exp (AIC (m)/n) are estimates of MSFE underhomoskedasticity
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Robust Mallows
C ∗n (m) = σ2(m) +2n
tr(Q(m)−1Ω(m)
)
Q(m) =1n
n−1∑t=0
xtx′t
Ω(m) =1n
n−1∑t=0
xtx′t e2t+1
where et+1 is residual from a preliminary estimate
An estimate of MSFE, robust to heteroskedasticity and serialcorrelation (similar to HAC standard errors)
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Cross-Validation
Leave-one-out estimator and prediction residual
β−t (m) =
(∑j 6=txj (m)xj (m)′
)−1 (∑j 6=txj (m)yj+1
)
et+1(m) = yt+1 − β−t (m)′xt (m)
et+1(m) is a forecast error based on estimation without observation t
E(et+1(m)2
)' MSFEn(m)
CVn(m) =1n
∑n−1t=0 et+1(m)
2 is an estimate of MSFEn(m)
Called the leave-one-out cross-validation (CV) criterion
Similar to Robust Mallows criterion
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Comments on CV Selection
Selecting one-step forecast models by cross-validation iscomputationally simple, generally valid, and robust toheteroskedasticity
Does not require correct specification
Similar to robust Mallows
Similar to Mallows, AIC and FPE under homoskedasticity
Conceptually easy to generalize beyond least-squares estimation
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Predictive Least Squares (Out-of-Sample MSFE)
Sequential estimates
βt (m) =
(t−1∑j=0xj (m)xj (m)′
)−1 (t−1∑j=0xj (m)yj+1
)
Sequential prediction residuals
et+1(m) = yt+1 − βt (m)′xt (m)
Predictive Least Squares. For some P
PLSn(m) =1P
n−1∑
t=n−Pet+1(m)2
Major Diffi culty: PLS very sensitive to P
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Comments on Predictive Least Squares
Conceptually simple, easy to generalize beyond least-squaresI Can be applied to actual forecasts, without need to know forecastmethod
et+1(m) are fully valid prediction errors
Possibly more robust to structural change than CVI Intuitive, but this claim has not been formally justified
Very common in applied forecastingI Frequently asserted as “empirical performance”
On the negative side, PLS over-estimates MSFEI et+1(m) is a prediction error from a sample of length t < nI PLS will tend to be overly-parsimoniousI Very sensitive to number of pseudo out-of-sample observations P
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Theory of Optimal Selection
MSFEn(m) is the MSFE from model m
infmMSFEn(m) is the (infeasible) best MSFE
Let m be the selected model
Let MSFEn(m) denote the MSFE using the selected estimator
We say that selection is asymptotically optimal if
MSFEn(m)infmMSFEn(m)
p−→ 1
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Theory of Optimal Selection
A series of papers have shown that AIC, Mallows, FPE areasymptotically optimal for selection
AssumptionsI AutoregressionsI Errors are iid, homoskedasticI True model is AR(∞)
Shibata (Annals, 1980), Ching-Kang Ing with co-authors (2003, 2005,etc)
Proof Method: Show that the selection criterion is uniformly close toMSFE
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Theory of Optimal Selection - Regression Case
In regression (iid date) case
Li (1987), Andrews (1991), Hansen (2007), Hansen and Racine(2012)
AIC, Mallows, FPE, CV are asymptotically optimal for seletion underhomoskedasticity
CV is asymptotically optimal for seletion under heteroskedasticity
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Forecast Selection - Summary
Testing inappropriate for forecast selection
Feasible selection criteria: BIC, AIC, AICc , Mallows, Robust Mallows,FPE, PLS, CV
Valid comparisons require holding sample constant across models
All methods except CV and PLS require conditional homoskedasticity
PLS sensitive to choice of P
BIC appropriate when true structure is sparse
CV quite general and flexibleI Recommended method
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GDP ExampleMethods: BIC, AICc , Robust Mallows, CV
Model BIC AICc C ∗n CVAR(1) 473 466 10.7 10.7AR(2) 472 462 10.6 10.5AR(3) 477 464 10.7 10.7AR(4) 481 465 10.8 10.8AR(5) 483 464 10.8 10.8AR(6) 489 466 11.0 10.9AR(7) 494 468 11.1 11.1AR(8) 498 470 11.3 11.2AR(9) 500 469 11.3 11.2AR(10) 505 471 11.4 11.4AR(11) 511 473 11.5 11.5AR(12) 511 471 11.4 11.3
Methods select AR(2)
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10-Year Treasury Rate
Model BIC AICc C ∗n CVAR(1) −1518 −1527 0.0798 0.0798AR(2) −1541∗ −1554 0.0768∗ 0.0768∗
AR(3) −1538 −1555 0.0769 0.0769AR(4) −1532 −1554 0.0773 0.0773AR(6) −1531 −1561 0.0772 0.0770AR(8) −1522 −1562 0.0777 0.0774AR(10) −1513 −1561 0.0784 0.0781AR(12) −1506 −1563 0.079 0.0787AR(20) −1471 −1561 0.081 0.080AR(22) −1470 −1570∗ 0.081 0.080AR(24) −1458 −1565 0.081 0.081
Mallows, AICc , FPE select AR(22)Robust Mallows, CV select AR(2)Difference due to conditional heteroskedasticityAR(2) through AR(6) near equivalent with respect to C ∗n and CVBruce Hansen (University of Wisconsin) Foundations October 29-31, 2013 56 / 108
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Point Forecast - GDP Growth
AR(2)
Data Forecast Actual2011:1 0.362011:2 1.332011:3 1.802011:4 2.912012:1 1.842012:2 2.65 1.20
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Point Forecast - 10-year Treasury Rate
AR(2)
Data Forecast ActualLevel Change Level Change Level Change
2012:1 1.97 -0.012012:2 1.97 0.002012:3 2.17 0.202012:4 2.05 -0.122012:5 1.96 -0.09 1.82 -0.23
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Forecasting with Leading Indicators
Recall, the ideal forecast is
E (yn+1|In) = E (yn+1|xn, xn−1, ...)
where In contains all information
xn = lags + leading indicatorsI Variables which help predict yt+1I We have focused on univariate lagsI Typically more information in related seriesI Which?
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Good Leading Indicators
Measured quickly
Anticipatory
Varies by forecast variable
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Interest Rate Spreads
Difference between Long and Short Rate
Measured immediately
Indicate monetary policy, aggregate demand
Term Structure of Interest Rates:
Long Rate is the market expectation of the average future short rates
Spread is the market expectation of future short rates
I use U.S. Treasury rates, difference between 10-year and 3-month
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Figure: 10-Year and 3-Month T-Bill Rates
1960 1970 1980 1990 2000 2010
24
68
1012
14
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Figure: Term Spread
1960 1970 1980 1990 2000 2010
10
12
34
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High Yield Spread
“Riskless” rate: U.S. Treasury
Low-risk rate: AAA grade corporate bond
High Yield rate: Low grade corporate bond
Theory: high-yield rate includes premium for probability of default
Low grade bond rates increase with probability of default —when realactivity is expected to fall
Spread: Difference between corporate bond rates
I use difference between AAA and BAA bond rates
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Figure: AAA and BAA Corporate Bond Rates
1960 1970 1980 1990 2000 2010
46
810
1214
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Figure: High Yield Spread
1960 1970 1980 1990 2000 2010
0.5
1.0
1.5
2.0
2.5
3.0
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Construction Indicators
Building Permits
Housing Starts
Anticipate construction spending
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Figure: Housing Starts, Building Permits
1960 1970 1980 1990 2000 2010
0.5
1.0
1.5
2.0
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Mixed Frequency Data
U.S. GDP is measured quarterly
Interest rates: Daily
Permits: Monthly
Simplest approach: Quarterly aggregationI Aggregate (average) daily and monthly variables to quarterly level
Mixed Frequency approachI Use lower frequency data as predictors
For now, we use aggregate (quarterly) data
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Timing
Variables reported in separate sequences
Should we use only "quarter 1" variables to forecast "quarter 2"?
Or should we use whatever is available?I E.g., use quarter 2 interest rates to forecast quarter 1 GDP?
Let’s use quarter 1 data to forecast quarter 2
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Models Selection by CVAll estimates include intercept plus two lags of GDP growth
Model CV ForecastSpread 10.4 2.8HY Spread 10.6 2.5Housing Starts 10.3 1.4Bulding Permits 10.3 1.7Sp+HY 10.3 2.7Sp+HS 10.02 1.5Sp+BP 10.1 1.9HY+HS 10.4 1.4HY+BP 10.4 1.6HS+BP 10.4 1.4Sp+HY+HS 10.00 1.3Sp+HY+BP 10.1 1.7Sp+HS+BP 10.05 1.3HY+HS+BP 10.5 1.3Sp+HY+HS+BP 10.02 1.1Bruce Hansen (University of Wisconsin) Foundations October 29-31, 2013 71 / 108
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CV-Selected Forecast: 1.3%Actual: 1.2%
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Coeffi cient Estimates
∆ log(GDPt+1) β s(β)Intercept −0.33 (1.03)∆ log(GDPt ) 0.16 (0.10)∆ log(GDPt−1) 0.09 (0.10)Bond Spreadt 0.61 (0.23)High Yield Spread −1.10 (0.75)Housing Startst 1.86 (0.65)
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Variance Forecasting
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Variance Forecasts
Forecast uncertaintyI Point forecasts insuffi cient!
σ2t+1 = var (yt+1|It )In the model yt+1 = β′xt + et+1
I σ2t+1 = var (et+1 |In) = E(e2t+1 |It
)
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10-Year Bond Rate
Prediction Residuals
Squares
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Figure: Leave-One-Out Prediction Residuals
1960 1970 1980 1990 2000 2010
1.5
1.0
0.5
0.0
0.5
1.0
1.5
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Figure: Squared Prediction Residuals
1960 1970 1980 1990 2000 2010
0.0
0.5
1.0
1.5
2.0
2.5
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Variance Forecast Methods
Constant Variance σ2t = σ2
I Uncertainty not state-dependent
GARCHI Common in financial dataI Estimated by MLE
Regression ApproachI σ2t = E
(e2t+1 |In
)≈ α′xt
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2-Step Variance Estimation
Start with residuals et+1I Better choice: leave-one-out residuals et+1
Estimate variance model (constant, ARCH, or regression)
Obtain σ2n from fitted model
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Which Residuals?
Least-squares residual variance biased toward zeroI Forecast variance biased towards zero
Leave-one-out residual variance estimates out-of-sample MSFEI Better choice
et+1(m) = yt+1 − β−t (m)′xt (m)
β−t (m) =
(∑j 6=txj (m)xj (m)′
)−1 (∑j 6=txj (m)yj+1
)
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Constant Variance Model
σ2t = σ2
σ2n = σ2 =1
n− 1 ∑n−1t=1 e
2t+1
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Regression Variance Model
σ2t ≈ α′xte2t+1 = α′xt + ηt
α =(∑n−1t=1 xtx
′t
)−1 (∑n−1t=1 xt e
2t+1
)σ2n = α′xn
I Easy, but not constrained to (0,∞)
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GARCH Models
σ2t = ω+ βσ2t−1 + αe2tConditional variance of et+1Specifies conditional variance as function of recent squaredinnovations
Large innovations (in magnitude) raise conditional variance
Lagged variance smooths σ2t
Non-negativity constraints: ω > 0, β ≥ 0, α > 0
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GARCH with Regressors
σ2t = ω+ βσ2t−1 + αe2t + γxtxt > 0 useful to constrain regressor to be positive
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Estimation by Quasi-LIkelihood
Numerical optimization
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Model Selection
Model with 2 ARCH lags and 2 regressors
σ2t = ω+ βσ2t−1 + α1e2t + α2e2t−1 + γ1x1t + γ2x2t
How many lags? How many regressors?
Presence of lagged σ2t−1 complicates issuesI β not identified when α1 = α2 = γ1 = γ2 = 0I This means conventional tests and information criterion are not correctwhen the process is close to constant variance
I We typically ignore this complication
Since estimation is nonlinear MLE much of model selection &combination literature is not relevant
I AIC appropriateI Unfortunately, not easy to compute with standard packages
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AIC for GARCH models
If model m has parameter vector θ(m) with k(m) elements
AIC (m) = 2L(θ(m)) + 2k(m)Not standard output
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Variance Forecast from GARCH model
σ2n+1 = ω+ βσ2n + α1e2nσ2n+1 = ω+ βσ2n + α1 e2nσ2n+1 is estimated conditional variance of yn+1
Standard deviation√
σ2n+1
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Example: 10-Year Bond Rate
GARCH(1,1)
σ2t = ω+ αe2t + βσ2t−1
Estimate s.e.ω 0.0001 0.0001α 0.200 0.041β 0.835 0.025
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Variance Forecast
Conditional varianceI σ2n+1 = 0.054I σn+1 = 0.23
UnconditionalI σ2 = 0.076I σ = 0.28
The conditional variance at present is similar, but somewhat smallerthan the unconditional
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Figure: Estimated Variance
1960 1970 1980 1990 2000 2010
0.0
0.2
0.4
0.6
0.8
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Example: GDP Growth
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Figure: GDP: Leave-One-Out Prediction Residuals
1970 1980 1990 2000 2010
10
50
510
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Figure: GDP: Squared Prediction Residuals
1970 1980 1990 2000 2010
050
100
150
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GARCH(1)σ2t = ω+ αe2t + βσ2t−1
Estimate s.e.ω 0.81 0.46α 0.21 0.06β 0.72 0.06
Conditional varianceI σ2n+1 = 4.1I σn+1 = 2.0
UnconditionalI σ2 = 9.8I σ = 3.1
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Figure: GDP: Estimated Variance
1970 1980 1990 2000 2010
1020
3040
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Interval Forecasting
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Interval Forecasts
Take the form [a, b]
Should contain yn+1 with probability 1− 2α
1− 2α = Pn (yn+1 ∈ [a, b])= Pn (yn+1 ≤ b)− Pn (yn+1 ≤ a)= Fn(b)− Fn(a)
where Fn(y) is the forecast distribution
It follows that
a = qn(α)
b = qn(1− α)
a = α’th and b = (1− α)’th quantile of conditional distribution
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Interval Forecasts are Conditional Quantiles
The ideal 80% forecast interval, is the 10% and 90% quantile of theconditional distribution of yn+1 given InOur feasible forecast intervals are estimates of the 10% and 90%quantile of the conditional distribution of yn+1 given InThe goal is to estimate conditional quantiles.
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Mean-Variance Model
Write
yt+1 = µt + σt εt+1
µt = E (yt+1|It )σ2t = var (yt+1|It )
Assume that εt+1 is independent of It .
Let qt (α) and qε(α) be the α’th quantiles of yt+1 and εt+1. Then
qt (α) = µt + σtqε(α)
Thus a (1− 2α) forecast interval for yn+1 is
[µn + σnqε(α), µn + σnqε(1− α)]
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Mean-Variance Model
Given the conditional mean µn and variance σ2n, the conditionalquantile of yn+1 is a linear function µn + σnqε(α) of the conditionalquantile qε(α) of the normalized error
εn+1 =en+1σn
Interval forecasts thus can be summarized by µn, σ2n, and qε(α)
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Normal Error Quantile Forecasts
Make the approximation εt+1 ∼ N(0, 1)I Then qε(α) = Z (a) are normal quantilesI Useful simplification, especially in small samples
0.10, 0.25, 0.75, 0.90 quantiles areI −1.285, −0.675, 0.675, 1.285
Forecast intervals
[µn + σnZ (α), µn + σnZ (1− α)]
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Nonparametric Error Quantile Forecasts
Let εt+1 ∼ F be unknownI We can estimate qε(α) as the empirical quantiles of the residualsI Set
εt+1 =et+1σt
I Sort ε1, ..., εn .I qε(α) and qε(1− α) are the α’th and (1− α)’th percentiles
[µn + σn qε(α), µn + σn qε(1− α)]
Computationally simple
Reasonably accurate when n ≥ 100Allows asymmetric and fat-tailed error distributions
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Constant Variance Case
If σt = σ is a constant, there is no advantage for estimation of σ forforecast interval
Let qe (α) and qe (1− α) be the α’th and (1− α)’th percentiles oforiginal residuals et+1Forecast Interval:
[µn + qε(α), µn + q
e (1− α)]
When the estimated variance is a constant, this is numericallyidentical to the definition with rescaled errors εt+1
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Example: Interest Rate Forecast
n = 603 observations
εt+1 =et+1σt
from GARCH(1,1) model
0.10, 0.25, 0.75, 0.90 quantiles
−1.16, −0.59, 0.62, 1.26Point Forecast = 1.96
50% Forecast interval = [1.82, 2.10]
80% Forecast interval = [1.69, 2.25]
Actual: 1.82
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Example: GDP
n = 207 observations
εt+1 =et+1σt
from GARCH(1,1) model
0.10, 0.25, 0.75, 0.90 quantiles
−1.18, −0.63, 0.57, 1.26Point Forecast = 1.31
50% Forecast interval = [0.04, 2.4]
80% Forecast interval = [−1.07, 3.8]
Actual: 1.20
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Mean-Variance Model Interval Forecasts - Summary
The key is to break the distribution into the mean µt , variance σ2t andthe normalized error εt+1
yt+1 = µt + σt εt+1
Then the distribution of yn+1 is determined by µn, σ2n and thedistribution of εn+1
Each of these three components can be separately approximated andestimated
Typically, we put the most work into modeling (estimating) the meanµt
I The remainder is modeled more simplyI For macro forecasts, this reflects a belief (assumption?) that most ofthe predictability is in the mean, not the higher features.
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