ECON/FIN 250: Forecasting in Finance and Economics ...people.brandeis.edu/~pmherb/FIN250/_downloads/filtering.pdf · ECON/FIN 250: Forecasting in Finance and Economics: Section 3.2

ECON/FIN 250: Forecasting in Finance and Economics:Section 3.2 Filtering Time Series

Patrick Herb

Brandeis University

Spring 2016

Patrick Herb (Brandeis University) Filtering Time Series ECON/FIN 250: Spring 2016 1 / 39

Course Overview

1 General Filtering Thoughts

2 Two-Sided Filters

3 One-Sided Filters

4 Local Trends


General Filtering Thoughts

Time series tools that can:Estimate TrendsSmooth Out NoiseAdjust for SeasonalityMake ForecastsSeparate the Cycle from the Trend

Somewhat ad hocCommon, simple rules used in business forecastingRelated to technical analysis in finance


Two-Sided Filters


2 Two-Sided Filters

3 One-Sided Filters

4 Local Trends


Moving Average Filter

Uses Data Before & After time = tSmooths Data Using Future & Past InformationCan Choose Information SetCan Weight Observations EquallyCan Choose Alternative WeightsNot Useful for ForecastingCan Improve Understanding of the Data


Moving Average Filter

Equally Weighted

y∗t = 1

2m + 1

m∑j=−m

yt+j (1)

Alternative Weighting

y∗t =

m∑j=−m

ωjyt+j (2)

Weights should have the property:m∑

j=−mωj = 1 (3)


Global Surface Temperature Anomaly

−.5

0.5

1S

urfa

ce T

emp

(C)

Ana

mol

y

1850 1900 1950 2000 2050Year



Let’s use an equally weighted moving average filter to reduce some of thenoise. Try m = 2use temperaturetsset yeartssmooth ma tempma = tempa , window (2 1 2)tsline tempa tempma



−.5

0.5

1

1850 1900 1950 2000 2050Year

Surface Temp (C) Anamoly ma: x(t)= tempa: window(2 1 2)



Still pretty noisy. Try m = 5tssmooth ma temp_ma = tempa , window (5 1 5)tsline tempa temp_ma



−.5

0.5

1

1850 1900 1950 2000 2050Year

Surface Temp (C) Anamoly ma: x(t)= tempa: window(5 1 5)


Low Frequency Filters

Can be used to remove the trendAllows the trend to changeCommon Macroeconomic Approach

Ignore GrowthConcentrate on Business CycleOutput Gap


Low Frequency Filters

ExamplesHodrick / Prescott FilterBaxter / KingButterworth

Type: help filter


Hodrick / Prescott Filter

The HP Filter:

miny∗

t

T∑t=1

(yt − y∗t )2 + λ

T−1∑t=2

[(y∗t+1 − y∗

t )− (y∗t − y∗

t−1]2 (4)

This filter is approximately a two-way moving average with weights subjectto a damped harmonic

λ is a positive smoothing parameterSmaller λ, penalizes variability in the cyclical componentLarger λ, penalizes variability in the growth component



There is some debate on choosing values for λ. Here are some suggestedvalues√

λ = σ1/σ2

Quarterly Data: λ = 1600Annual Data: λ = 100Monthly Data: λ = 14400

For more, see the original paper posted on LatteHodrick, Robert J, and Edward C. Prescott. “Postwar U.S. BusinessCycles: An Empirical Investigation.” Journal of Money, Credit, andBanking. Vol. 29, No. 1, February 1997, pp. 1-16.



The Model:

log(GDP) = Trend + Cycle (5)= Growth + Cycle (6)

HP Filter in Statause gdptsfilter hp gdpcycle = lgdp , trend( gdptrend )

Note: λ = 1600 is default value


Log GDP & HP Trend

7.5

88.

59

9.5

1947q3 1964q3 1981q3 1998q3 2015q3dateq

lgdp lgdp trend component from hp filter


Log GDP & HP Trend

9.4

9.5

9.6

9.7

2000q1 2005q1 2010q1 2015q1dateq

lgdp lgdp trend component from hp filter

tsline lgdp gdptrend if dateq > tq (2000q1)


HP Cycle

−.0

6−

.04

−.0

20

.02

.04

lgdp

cyc

lical

com

pone

nt fr

om h

p fil

ter

1947q3 1964q3 1981q3 1998q3 2015q3dateq


Business Cycles & HP Filter

Much of modern macroeconomics applies the HP filter to many series toseparate the trend from the cycle:

GDPConsumptionInvestment

Then looks at the comovement between the cyclical components


One-Sided Filters


2 Two-Sided Filters

3 One-Sided Filters

4 Local Trends


Moving Average Again

y∗t = 1

m

1∑j=m

yt−j (7)

use unemploytssmooth ma unma = unrate , window (5 ,0 ,0)

y∗t = 1

5

5∑j=1

yt−j (8)


Exponential Weighted Moving Average (EWMA)

Average of past valuesDecreasing (exponentially) weightsRelated to many ideas / models in economics

Adaptive Expectations (Friedman)Rational ExpectationsAutoregressive ModelsRiskmetricsGARCH Models



EWMAy∗

t = αyt + (1− α)y∗t−1 (9)

Recursive substitution reveals declining weights

y∗t = αyt + (1− α)(αyt−1 + (1− α)y∗

t−2)y∗

t = αyt + α(1− α)yt−1 + (1− α)2y∗t−2

...y∗

t = αyt + α(1− α)y∗t−1 + α(1− α)2yt−2 + ...+ (1− α)ty∗

0 (10)



2 4 6 8 10 12 14 16 18 20

yt-j

0.05

0.1

0.15

0.2

0.25

0.3W

eigh

t

α = 0.3α = 0.2α = 0.1



Estimate the unemployment rate trenduse unemploytssmooth exp unewma = unrate , parms (0.1)

This sets α = 0.1Stata can find the optimal α by excluding the parms() options

Be careful with this!



24

68

10

1950m1 1960m1 1970m1 1980m1 1990m1 2000m1 2010m1 2020m1datem

unrate exp parms(0.1000) = unrate


The Optimality of the EWMA

Suppose your time series comes from the following process:

xt = xt−1 + et xt = random walkyt = xt + ηt et , ηt = white noise

You do not observe xt

This implies yt is a random walk plus noiseIn this case, EWMA is the optimal forecast


Local Trends


2 Two-Sided Filters

3 One-Sided Filters

4 Local Trends


Double Exponential Weighted Moving Average (DEWMA)

Filter Oncey∗

t = αyt + (1− α)y∗t−1 (11)

Filter Againy∗∗

t = αy∗t + (1− α)y∗∗

t−1 (12)

The double filter can follow local trendsThe trend can keep going up when the observed yt is going downGetting initial values for y∗

0 , y∗∗0 can be tricky


Double Exponential Weighted Moving Average

Estimate the unemployment rate trend using the double exponentialweighted moving averageuse unemploytssmooth dexp unema = unrate , parms (0.1)tsline unrate unema


Double Exponential Weighted Moving Average

24

68

10

1950m1 1960m1 1970m1 1980m1 1990m1 2000m1 2010m1 2020m1datem

unrate dexp parms(0.1000) = unrate


Holt-Winters Smoother

y∗t = at−1 + bt−1 (13)

at = αyt + (1− α)(at−1 + bt−1) (14)bt = β(at − at−1) + (1− β)bt−1 (15)


Holt-Winters Smoother

Estimate the unemployment rate trend using the Holt-Winters smootherwith parameters α = 0.1, β = 0.2use unemploytssmooth hwinters unhw = unrate , parms (0.1 0.2)tsline unrate unhw


Holt-Winters & Unemployment Rate

24

68

1012

1950m1 1960m1 1970m1 1980m1 1990m1 2000m1 2010m1 2020m1datem

unrate hw parms(0.100 0.200) = unrate


Holt-Winters Seasonal Smoother

Seasonal version of Holt-WintersSt is a repeating seasonal adjustmentThis is the multiplicative version (there is an additive version)s specifies the period of the seasonality

monthly data would have s = 12

y∗t = (at−1 + bt−1)St (16)

at = αyt

St−s) + (1− α)(at−1 + bt−1) (17)

bt = β(at − at−1) + (1− β)bt−1 (18)

St = γytat

+ (1− γ)St−s (19)


Holt-Winters Seasonal Smoother

Estimate the unemployment rate trend using:Seasonal Holt-Winters smootherNot seasonally adjusted unemployment rate monthly dataChoose parameters α = 0.1, β = 0.2, γ = 0.05

use unemploy

tssmooth shwinters unhws = unratensa , //parms (0.1 0.2 0.05) period (12)

tsline unratensa unhws if datem > tm (1990m1)


Seasonal Holt-Winters & Unemployment Rate NSA

46

810

12

1995m1 2000m1 2005m1 2010m1 2015m1datem

unratensa shw parms(0.100 0.200 0.050) = unratensa


Summary

yt = (Trend) + (Seasonal) + (Cycle) + (Noise) (20)

Filters can help separate partsFilters can help smooth out the noiseNot necessarily predictiveSometimes difficult to estimate

Initial value problemsAlso, a little ad hoc

Not clear what the data generating process is


Documents

ECON/FIN 250: Forecasting in Finance and Economics ...people.brandeis.edu/~pmherb/FIN250/_downloads/filtering.pdf · ECON/FIN 250: Forecasting in Finance and Economics: Section 3.2