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

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1 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 / 39

2 Course Overview 1 General Filtering Thoughts 2 Two-Sided Filters 3 One-Sided Filters 4 Local Trends Patrick Herb (Brandeis University) Filtering Time Series ECON/FIN 250: Spring / 39

3 General Filtering Thoughts Time series tools that can: Estimate Trends Smooth Out Noise Adjust for Seasonality Make Forecasts Separate the Cycle from the Trend Somewhat ad hoc Common, simple rules used in business forecasting Related to technical analysis in finance Patrick Herb (Brandeis University) Filtering Time Series ECON/FIN 250: Spring / 39

4 Two-Sided Filters 1 General Filtering Thoughts 2 Two-Sided Filters 3 One-Sided Filters 4 Local Trends Patrick Herb (Brandeis University) Filtering Time Series ECON/FIN 250: Spring / 39

5 Moving Average Filter Uses Data Before & After time = t Smooths Data Using Future & Past Information Can Choose Information Set Can Weight Observations Equally Can Choose Alternative Weights Not Useful for Forecasting Can Improve Understanding of the Data Patrick Herb (Brandeis University) Filtering Time Series ECON/FIN 250: Spring / 39

6 Moving Average Filter Equally Weighted Alternative Weighting y t = 1 2m + 1 y t = Weights should have the property: m j= m m j= m m j= m y t+j (1) ω j y t+j (2) ω j = 1 (3) Patrick Herb (Brandeis University) Filtering Time Series ECON/FIN 250: Spring / 39

7 Global Surface Temperature Anomaly Surface Temp (C) Anamoly Year Patrick Herb (Brandeis University) Filtering Time Series ECON/FIN 250: Spring / 39

8 Global Surface Temperature Anomaly Let s use an equally weighted moving average filter to reduce some of the noise. Try m = 2 use temperature tsset year tssmooth ma tempma = tempa, window (2 1 2) tsline tempa tempma Patrick Herb (Brandeis University) Filtering Time Series ECON/FIN 250: Spring / 39

9 Global Surface Temperature Anomaly Year Surface Temp (C) Anamoly ma: x(t)= tempa: window(2 1 2) Patrick Herb (Brandeis University) Filtering Time Series ECON/FIN 250: Spring / 39

10 Global Surface Temperature Anomaly Still pretty noisy. Try m = 5 tssmooth ma temp _ ma = tempa, window (5 1 5) tsline tempa temp _ ma Patrick Herb (Brandeis University) Filtering Time Series ECON/FIN 250: Spring / 39

11 Global Surface Temperature Anomaly Year Surface Temp (C) Anamoly ma: x(t)= tempa: window(5 1 5) Patrick Herb (Brandeis University) Filtering Time Series ECON/FIN 250: Spring / 39

12 Low Frequency Filters Can be used to remove the trend Allows the trend to change Common Macroeconomic Approach Ignore Growth Concentrate on Business Cycle Output Gap Patrick Herb (Brandeis University) Filtering Time Series ECON/FIN 250: Spring / 39

13 Low Frequency Filters Examples Hodrick / Prescott Filter Baxter / King Butterworth Type: help filter Patrick Herb (Brandeis University) Filtering Time Series ECON/FIN 250: Spring / 39

14 Hodrick / Prescott Filter The HP Filter: min y t T T (y t yt ) λ [(yt+1 yt ) (yt yt 1] 2 (4) t=1 t=2 This filter is approximately a two-way moving average with weights subject to a damped harmonic λ is a positive smoothing parameter Smaller λ, penalizes variability in the cyclical component Larger λ, penalizes variability in the growth component Patrick Herb (Brandeis University) Filtering Time Series ECON/FIN 250: Spring / 39

15 Hodrick / Prescott Filter There is some debate on choosing values for λ. Here are some suggested values λ = σ1 /σ 2 Quarterly Data: λ = 1600 Annual Data: λ = 100 Monthly Data: λ = For more, see the original paper posted on Latte Hodrick, Robert J, and Edward C. Prescott. Postwar U.S. Business Cycles: An Empirical Investigation. Journal of Money, Credit, and Banking. Vol. 29, No. 1, February 1997, pp Patrick Herb (Brandeis University) Filtering Time Series ECON/FIN 250: Spring / 39

16 Hodrick / Prescott Filter The Model: log(gdp) = Trend + Cycle (5) = Growth + Cycle (6) HP Filter in Stata use gdp tsfilter hp gdpcycle = lgdp, trend ( gdptrend ) Note: λ = 1600 is default value Patrick Herb (Brandeis University) Filtering Time Series ECON/FIN 250: Spring / 39

17 Log GDP & HP Trend q3 1964q3 1981q3 1998q3 2015q3 dateq lgdp lgdp trend component from hp filter Patrick Herb (Brandeis University) Filtering Time Series ECON/FIN 250: Spring / 39

18 Log GDP & HP Trend q1 2005q1 2010q1 2015q1 dateq lgdp lgdp trend component from hp filter tsline lgdp gdptrend if dateq > tq ( 2000 q 1) Patrick Herb (Brandeis University) Filtering Time Series ECON/FIN 250: Spring / 39

19 HP Cycle lgdp cyclical component from hp filter q3 1964q3 1981q3 1998q3 2015q3 dateq Patrick Herb (Brandeis University) Filtering Time Series ECON/FIN 250: Spring / 39

20 Business Cycles & HP Filter Much of modern macroeconomics applies the HP filter to many series to separate the trend from the cycle: GDP Consumption Investment Then looks at the comovement between the cyclical components Patrick Herb (Brandeis University) Filtering Time Series ECON/FIN 250: Spring / 39

21 One-Sided Filters 1 General Filtering Thoughts 2 Two-Sided Filters 3 One-Sided Filters 4 Local Trends Patrick Herb (Brandeis University) Filtering Time Series ECON/FIN 250: Spring / 39

22 Moving Average Again yt = 1 1 y t j (7) m j=m use unemploy tssmooth ma unma = unrate, window (5,0,0) yt = 1 5 y t j (8) 5 j=1 Patrick Herb (Brandeis University) Filtering Time Series ECON/FIN 250: Spring / 39

23 Exponential Weighted Moving Average (EWMA) Average of past values Decreasing (exponentially) weights Related to many ideas / models in economics Adaptive Expectations (Friedman) Rational Expectations Autoregressive Models Riskmetrics GARCH Models Patrick Herb (Brandeis University) Filtering Time Series ECON/FIN 250: Spring / 39

24 Exponential Weighted Moving Average (EWMA) EWMA yt = αy t + (1 α)yt 1 (9) Recursive substitution reveals declining weights yt = αy t + (1 α)(αy t 1 + (1 α)yt 2) yt = αy t + α(1 α)y t 1 + (1 α) 2 yt 2. yt = αy t + α(1 α)yt 1 + α(1 α) 2 y t (1 α) t y0 (10) Patrick Herb (Brandeis University) Filtering Time Series ECON/FIN 250: Spring / 39

25 Exponential Weighted Moving Average (EWMA) α = 0.3 α = 0.2 α = Weight y t-j Patrick Herb (Brandeis University) Filtering Time Series ECON/FIN 250: Spring / 39

26 Exponential Weighted Moving Average (EWMA) Estimate the unemployment rate trend use unemploy tssmooth exp unewma = unrate, parms ( 0. 1) This sets α = 0.1 Stata can find the optimal α by excluding the parms() options Be careful with this! Patrick Herb (Brandeis University) Filtering Time Series ECON/FIN 250: Spring / 39

27 Exponential Weighted Moving Average (EWMA) m1 1960m1 1970m1 1980m1 1990m1 2000m1 2010m1 2020m1 datem unrate exp parms(0.1000) = unrate Patrick Herb (Brandeis University) Filtering Time Series ECON/FIN 250: Spring / 39

28 The Optimality of the EWMA Suppose your time series comes from the following process: x t = x t 1 + e t y t = x t + η t x t = random walk e t, η t = white noise You do not observe x t This implies y t is a random walk plus noise In this case, EWMA is the optimal forecast Patrick Herb (Brandeis University) Filtering Time Series ECON/FIN 250: Spring / 39

29 Local Trends 1 General Filtering Thoughts 2 Two-Sided Filters 3 One-Sided Filters 4 Local Trends Patrick Herb (Brandeis University) Filtering Time Series ECON/FIN 250: Spring / 39

30 Double Exponential Weighted Moving Average (DEWMA) Filter Once Filter Again y t = αy t + (1 α)y t 1 (11) y t = αy t + (1 α)y t 1 (12) The double filter can follow local trends The trend can keep going up when the observed y t is going down Getting initial values for y 0, y 0 can be tricky Patrick Herb (Brandeis University) Filtering Time Series ECON/FIN 250: Spring / 39

31 Double Exponential Weighted Moving Average Estimate the unemployment rate trend using the double exponential weighted moving average use unemploy tssmooth dexp unema = unrate, parms ( 0. 1) tsline unrate unema Patrick Herb (Brandeis University) Filtering Time Series ECON/FIN 250: Spring / 39

32 Double Exponential Weighted Moving Average m1 1960m1 1970m1 1980m1 1990m1 2000m1 2010m1 2020m1 datem unrate dexp parms(0.1000) = unrate Patrick Herb (Brandeis University) Filtering Time Series ECON/FIN 250: Spring / 39

33 Holt-Winters Smoother y t = a t 1 + b t 1 (13) a t = αy t + (1 α)(a t 1 + b t 1 ) (14) b t = β(a t a t 1 ) + (1 β)b t 1 (15) Patrick Herb (Brandeis University) Filtering Time Series ECON/FIN 250: Spring / 39

34 Holt-Winters Smoother Estimate the unemployment rate trend using the Holt-Winters smoother with parameters α = 0.1, β = 0.2 use unemploy tssmooth hwinters unhw = unrate, parms ( ) tsline unrate unhw Patrick Herb (Brandeis University) Filtering Time Series ECON/FIN 250: Spring / 39

35 Holt-Winters & Unemployment Rate m1 1960m1 1970m1 1980m1 1990m1 2000m1 2010m1 2020m1 datem unrate hw parms( ) = unrate Patrick Herb (Brandeis University) Filtering Time Series ECON/FIN 250: Spring / 39

36 Holt-Winters Seasonal Smoother Seasonal version of Holt-Winters S t is a repeating seasonal adjustment This is the multiplicative version (there is an additive version) s specifies the period of the seasonality monthly data would have s = 12 y t = (a t 1 + b t 1 )S t (16) a t = α y t S t s ) + (1 α)(a t 1 + b t 1 ) (17) b t = β(a t a t 1 ) + (1 β)b t 1 (18) S t = γ y t a t + (1 γ)s t s (19) Patrick Herb (Brandeis University) Filtering Time Series ECON/FIN 250: Spring / 39

37 Holt-Winters Seasonal Smoother Estimate the unemployment rate trend using: Seasonal Holt-Winters smoother Not seasonally adjusted unemployment rate monthly data Choose parameters α = 0.1, β = 0.2, γ = 0.05 use unemploy tssmooth shwinters unhws = unratensa, // parms ( ) period (12) tsline unratensa unhws if datem > tm ( 1990 m 1) Patrick Herb (Brandeis University) Filtering Time Series ECON/FIN 250: Spring / 39

38 Seasonal Holt-Winters & Unemployment Rate NSA m1 2000m1 2005m1 2010m1 2015m1 datem unratensa shw parms( ) = unratensa Patrick Herb (Brandeis University) Filtering Time Series ECON/FIN 250: Spring / 39

39 Summary y t = (Trend) + (Seasonal) + (Cycle) + (Noise) (20) Filters can help separate parts Filters can help smooth out the noise Not necessarily predictive Sometimes difficult to estimate Initial value problems Also, a little ad hoc Not clear what the data generating process is Patrick Herb (Brandeis University) Filtering Time Series ECON/FIN 250: Spring / 39