ECON/FIN 250: Forecasting in Finance and Economics: Section 8: Forecast Examples: Part 1

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1 ECON/FIN 250: Forecasting in Finance and Economics: Section 8: Forecast Examples: Part 1 Patrick Herb Brandeis University Spring 2016 Patrick Herb (Brandeis University) Forecast Examples: Part 1 ECON/FIN 250: Spring / 40

2 Course Overview 1 Key Objectives 2 The Forecast Process 3 US GDP: Unit Root 4 Multi-Period Forecasts Patrick Herb (Brandeis University) Forecast Examples: Part 1 ECON/FIN 250: Spring / 40

3 Key Objectives 1 Key Objectives 2 The Forecast Process 3 US GDP: Unit Root 4 Multi-Period Forecasts Patrick Herb (Brandeis University) Forecast Examples: Part 1 ECON/FIN 250: Spring / 40

4 Key Objectives The Forecast Process US GDP: Unit Root Multi-Step Forecasts US GDP: Time Trend US Unemployment Gasoline Supply ARMAX Rolling & Recursive Forecasts Patrick Herb (Brandeis University) Forecast Examples: Part 1 ECON/FIN 250: Spring / 40

5 The Forecast Process 1 Key Objectives 2 The Forecast Process 3 US GDP: Unit Root 4 Multi-Period Forecasts Patrick Herb (Brandeis University) Forecast Examples: Part 1 ECON/FIN 250: Spring / 40

6 The Forecast Process Model Identification Residual Diagnostics Single & Multi-Step Forecasts Forecast Confidence Intervals Patrick Herb (Brandeis University) Forecast Examples: Part 1 ECON/FIN 250: Spring / 40

7 Examples US GDP US Unemployment US Gasoline Supply Patrick Herb (Brandeis University) Forecast Examples: Part 1 ECON/FIN 250: Spring / 40

8 US GDP: Unit Root 1 Key Objectives 2 The Forecast Process 3 US GDP: Unit Root 4 Multi-Period Forecasts Patrick Herb (Brandeis University) Forecast Examples: Part 1 ECON/FIN 250: Spring / 40

9 What Have We Done So Far? Looked at GDP visually with tsline; Log-linear Looked at ACF and PACF of lgdp Estimated AR(1) and AR(1/2); roots looked unstable Unit Root Test: Dickey Fuller (which flavor?); Concluded lgdp has unit root Now what? Patrick Herb (Brandeis University) Forecast Examples: Part 1 ECON/FIN 250: Spring / 40

10 Integrating Log Real GDP To integrate log real GDP, take the first difference y t = log(gdp t ) log(gdp t 1 ) (1) Model y t as an ARMA(p,q) or ARIMA(p,1,q) In words: US GDP growth rates follow ARMA We will still check time trend models too Patrick Herb (Brandeis University) Forecast Examples: Part 1 ECON/FIN 250: Spring / 40

11 Identifying the Process for Log Real GDP (y t ) In Stata: gen dgdp=d.lgdp and plot Check ACF and PACF; Some Correlation Model as AR(1), AR(2), MA(1), MA(2), ARMA(1,1); check roots Slight preference to AR models; simpler to estimate We will formalize some model selection criterion Patrick Herb (Brandeis University) Forecast Examples: Part 1 ECON/FIN 250: Spring / 40

12 Fitting AR Models AR models are easy to estimate Can use OLS In Stata use Lag operator This brings up a few theoretical issues Patrick Herb (Brandeis University) Forecast Examples: Part 1 ECON/FIN 250: Spring / 40

13 In Stata * estimate AR (2) with OLS reg dgdp L. dgdp L2. dgdp * now with arima command arima dgdp, ar (1/2) * MA (2) with arima arima dgdp, ma (2) * can also estimate fancy pattern arima dgdp, ar (1 3 5) * can estimate the full model from lgdp arima lgdp, arima (p,1,q) Patrick Herb (Brandeis University) Forecast Examples: Part 1 ECON/FIN 250: Spring / 40

14 OLS & Lagged Dependent Variables Lagged dependent variables: y t = β 0 + αy t 1 + β 1 x t + η t η t is iid (independently and identically distributed) η t is weakly exogenous, E[η s x t, y t 1 ] = 0, s t OLS Properties Parameter estimates biased, consistent Asymptotic distributions and standard errors/tests correct Many time series models will be like this Patrick Herb (Brandeis University) Forecast Examples: Part 1 ECON/FIN 250: Spring / 40

15 Fitting ARMA Models These use numerical optimization procedures Computer searches for BEST parameters Maximum Likelihood Estimation (MLE) Patrick Herb (Brandeis University) Forecast Examples: Part 1 ECON/FIN 250: Spring / 40

16 What are Likelihood Functions? Estimate the probability of data, given parameters Maximize Prob(Data Parameters) Computer over parameters and finds most likely choice Patrick Herb (Brandeis University) Forecast Examples: Part 1 ECON/FIN 250: Spring / 40

17 Estimating the Likelihood Estimated disturbances (ARMA(1,1)): ê t = y t φy t 1 θê t 1, σ 2 = Var[ê t ] (2) T T T 1 f t = f (ê t φ, θ, σ) = t=1 t=1 t=1 2πσ 2 exp[ ê2 t 2σ ] (3) 2 L = log( f t ) = 1 T T log(2πσ 2 êt 2 ) (4) 2 t=1 t=1 2σ 2 L = T 2 log(2πσ2 ) L = T 2 log(2πσ2 ) T t=1 ê 2 t 2σ 2 (5) Tt=1 ê 2 t 2σ 2 (6) Patrick Herb (Brandeis University) Forecast Examples: Part 1 ECON/FIN 250: Spring / 40

18 Identification Methods Try Different Models Look at Parameter t-statistics (Standard Errors) Look at Residuals (Correlations and Q-Test) Look at Information Criterion Patrick Herb (Brandeis University) Forecast Examples: Part 1 ECON/FIN 250: Spring / 40

19 Portmanteau Q Test Model Residuals (where x t is vector of ARMA terms) ɛ t = y t x t β (7) Estimate autocorrelations, ˆρ j for L lags, for data with sample size T. (Note, these are NOT partial autocorrelations) Estimate the sum of the squared autocorrelations with fancy weights Q = T (T 2) 1 L 1 L T i ˆρ2 i (8) For white noise: Q χ 2 L Reject white noise (in favor of serial correlation) if Q > χ 2 L i=1 Patrick Herb (Brandeis University) Forecast Examples: Part 1 ECON/FIN 250: Spring / 40

20 Portmanteau Q Test * estimate arima (2,1,0) for log gdp arima lgdp, arima (2,1,0) * create residuals predict resid, res * test for white noise at 20 lags wntestq resid, lags ( 20) * test for white noise at ( this case ) 40 lags wntestq resid Patrick Herb (Brandeis University) Forecast Examples: Part 1 ECON/FIN 250: Spring / 40

21 Information Criterion Akaike Information Criterion (AIC) AIC = 2L + 2k (9) Bayesian Information Criterion (BIC) BIC = 2L + k log(t ) (10) Choose model with smallest AIC or BIC Computer will attempt to make L big The number of parameters, k, is a penalty More parameters result in larger AIC & BIC Patrick Herb (Brandeis University) Forecast Examples: Part 1 ECON/FIN 250: Spring / 40

22 Stata Diagnostics Compare 2 models * model 1 arima lgdp, arima (1,1,0) predict res 1, res estat aroots wntestq res 1, lags (10) estat ic * model 2 arima lgdp, arima (2,1,0) predict res 2, res estat aroots wntestq res 2, lags (10) estat ic Patrick Herb (Brandeis University) Forecast Examples: Part 1 ECON/FIN 250: Spring / 40

23 One-Step Ahead Forecasts * one - step ahead fcast for gdp growth rate arima dgdp, ar (1) predict yhat * compare growth rate to fcast scatter dgdp yhat tsline dgdp yhat * one - ste ahead fcast for GDP level arima lgdp, arima (1,1,0) predict fcast, y * compare to naive forecast gen naive = lgdp [_ n -1] tsline lgdp fcast naive if dateq > tq ( 2003 q 4) Patrick Herb (Brandeis University) Forecast Examples: Part 1 ECON/FIN 250: Spring / 40

24 One-Step Ahead Forecast Comparison q1 2007q1 2010q1 2013q1 2016q1 dateq lgdp naive y prediction, one step Patrick Herb (Brandeis University) Forecast Examples: Part 1 ECON/FIN 250: Spring / 40

25 95% Confidence Bands: Example AR(1) E[(y t+1 ŷ t+1 ) 2 ], [ŷ t ˆσ ɛ, ŷ t ˆσ ɛ ] (11) y t+1 = ρy t + ɛ t (12) ŷ t+1 = ˆρy t (13) y t+1 = ˆρy t + ˆɛ t+1 (14) E[(y t+1 ŷ t+1 ) 2 ] = E[(y t+1 ˆρy t ) 2 ] (15) = E[ɛ 2 t+1] (16) = σ ɛ (17) Patrick Herb (Brandeis University) Forecast Examples: Part 1 ECON/FIN 250: Spring / 40

26 95% Confidence Bands: Example AR(1) Two Parts Don t know ɛ t+1 or σ ɛ Don t know true ρ Forecast uncertainty σŷ We will often ignore forecast uncertainty (smaller, more difficult) Patrick Herb (Brandeis University) Forecast Examples: Part 1 ECON/FIN 250: Spring / 40

27 Confidence Bands: log(gdp) ARIMA(1,1,0) y t = log(gdp t ) (18) y t+1 = y t + x t+1 (19) x t+1 = a + ρx t + ɛ t+1 (20) E[(x t+1 E[x t+1 Ω t ]) 2 ] = E[(x t+1 (a + ρx t )) 2 ] (21) = E[ɛ 2 t ] (22) = σ 2 ɛ (23) [ŷ t ˆσ ɛ, ŷ t ˆσ ɛ ] (24) Patrick Herb (Brandeis University) Forecast Examples: Part 1 ECON/FIN 250: Spring / 40

28 Confidence Bands: log(gdp) arima lgdp, arima (1,1,0) predict yhat, y predict sigma 2, mse gen sigma = sqrt ( sigma 2) gen yl = yhat * sigma gen yh = yhat * sigma tsline lgdp yhat yl yh if dateq > tq ( 2003 q 4) Patrick Herb (Brandeis University) Forecast Examples: Part 1 ECON/FIN 250: Spring / 40

29 Confidence Bands: log(gdp) q1 2007q1 2010q1 2013q1 2016q1 dateq lgdp yl y prediction, one step yh Patrick Herb (Brandeis University) Forecast Examples: Part 1 ECON/FIN 250: Spring / 40

30 In/Out of Sample Forecasts Recall the first days of our class? In sample forecasts can look too good Need to check out of sample Let s try this with GDP Easy to do with Stata Patrick Herb (Brandeis University) Forecast Examples: Part 1 ECON/FIN 250: Spring / 40

31 In/Out of Sample Forecast Comparison * we want to compare mse of in/ out sample use gdp * model 1: in sample arima lgdp, arima (1,1,0) predict fcast 1, y gen in_e2 = (lgdp - fcast 1)ˆ2 if dateq >= tq (2000 q1) * model 2: out of sample arima lgdp if dateq <tq (2000 q1), arima (1,1,0) predict fcast 2, y gen out _e2 = (lgdp - fcast 2)ˆ2 if dateq >= tq (2000 q1) * compare mse sum in_e2 out _e2 Patrick Herb (Brandeis University) Forecast Examples: Part 1 ECON/FIN 250: Spring / 40

32 Out of Sample Tests Need to be able to compare out of sample forecasting performance Choosing in and out of sample periods is difficult Large in sample estimation Good parameter estimates (in sample) Noisy MSE estimate (out of sample) Small in sample estimation Noisy parameter estimates (in sample) Good MSE estimate (out of sample) Patrick Herb (Brandeis University) Forecast Examples: Part 1 ECON/FIN 250: Spring / 40

33 Multi-Period Forecasts 1 Key Objectives 2 The Forecast Process 3 US GDP: Unit Root 4 Multi-Period Forecasts Patrick Herb (Brandeis University) Forecast Examples: Part 1 ECON/FIN 250: Spring / 40

34 Two Flavors of Multi-Period Forecasts Direct: Build forecast for y t+h from t Recursive: Keep substituting one-step ahead forecasts into your equations Patrick Herb (Brandeis University) Forecast Examples: Part 1 ECON/FIN 250: Spring / 40

35 Recursive Multi-Period Forecasts Go more than one period into the future Replace endogenous variables with forecasts Example: AR(1) Remember all of our expected value theory from the last few classes? y t+1 = ρy t + ɛ t (25) ŷ t+1 = ˆρy t (26) ŷ t+2 = ˆρŷ t+1 = ˆρ 2 y t (27) ŷ t+3 = ˆρŷ t+2 = ˆρ 3 y t (28) Patrick Herb (Brandeis University) Forecast Examples: Part 1 ECON/FIN 250: Spring / 40

36 Stata for Multi-Period Forecasts Dynamic option only works after an appropriate arima estimation dynamic(tq(2010q1)) starts the true forecast q Questions? type help arima postestimation use gdp arima lgdp if dateq <= tq (2009 q4), arima (1,1,0) predict yhat, y dynamic (tq (2010 q 1)) tsline lgdp yhat if dateq > tq ( 2000 q 4) Patrick Herb (Brandeis University) Forecast Examples: Part 1 ECON/FIN 250: Spring / 40

37 Multi-Period Forecast q1 2005q1 2010q1 2015q1 dateq lgdp y prediction, dyn(tq(2009q4)) Patrick Herb (Brandeis University) Forecast Examples: Part 1 ECON/FIN 250: Spring / 40

38 Multi-Step Confidence Bands y t+1 = ρy t + ɛ t+1 y t+2 = ρy t+1 + ɛ t+2 = ρ 2 y t + ρɛ t+1 + ɛ t+2 E(y t+2 ŷ t+2 ) 2 = ρ 2 σ 2 ɛ + σ 2 ɛ For big h, E(y t+h ŷ t+h ) 2 = σ 2 ɛ h 1 i=0 (ρ 2 ) i E(y t+h ŷ t+h ) 2 = σ 2 ɛ (ρ 2 ) i = 1 i=0 1 ρ 2 σ2 ɛ = Var[y t ] Patrick Herb (Brandeis University) Forecast Examples: Part 1 ECON/FIN 250: Spring / 40

39 Finally: A Real Forecast What does our model say about future GDP? How would you produce actual forecasts? Need tsappend use gdp * use full sample arima lgdp, arima (1,1,0) * add 8 quarters to the data set tsappend, add (8) * try without dynamic predict yhat 1, y * now try dynamic with date predict yhat, y dynamic (tq (2016 q 1)) tsline lgdp yhat if dateq > tq ( 2001 q 4) Patrick Herb (Brandeis University) Forecast Examples: Part 1 ECON/FIN 250: Spring / 40

40 Real Forecast of log(gdp) q1 2006q1 2010q1 2014q1 2018q1 dateq lgdp y prediction, dyn(tq(2016q1)) Patrick Herb (Brandeis University) Forecast Examples: Part 1 ECON/FIN 250: Spring / 40

ECON/FIN 250: Forecasting in Finance and Economics: Section 7: Unit Roots & Dickey-Fuller Tests

ECON/FIN 250: Forecasting in Finance and Economics: Section 7: Unit Roots & Dickey-Fuller Tests Patrick Herb Brandeis University Spring 2016 Patrick Herb (Brandeis University) Unit Root Tests ECON/FIN