INTRODUCTION TO TIME SERIES ANALYSIS. The Simple Moving Average Model

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1 INTRODUCTION TO TIME SERIES ANALYSIS The Simple Moving Average Model

2 The Simple Moving Average Model The simple moving average (MA) model: More formally: where t is mean zero white noise (WN). Three parameters: The mean µ The slope The WN variance

3 MA Processes - I If slope is zero then: Y t = µ + t And Y is White Noise ( µ, 2 ) t

4 MA Processes - II If slope is not zero then depends on both Y t t and And the process { Y t } is autocorrelated Large values of lead to greater autocorrelation Negative values of result in oscillatory time series

5 MA Examples θ = 0.5 θ = 0.9 Y t Y t Time (t) Time (t) θ = 0.5 θ = 0 Y t Y t Time (t) Time (t)

6 Autocorrelations ρ(h) θ = 0.75 ρ(h) θ = 0.5 Only lag 1 autocorrelation non-zero for the MA model h h ρ(h) θ = 0.2 ρ(h) θ = h h

7 INTRODUCTION TO TIME SERIES ANALYSIS Let s practice!

8 INTRODUCTION TO TIME SERIES ANALYSIS MA Model Estimation and Forecasting

9 MA Processes: Changes in Inflation Rate - I One-month US inflation rate (in percent, annual rate) Monthly observations from 1950 through 1990 > data(mishkin, package = "Ecdat") > inflation <- as.ts(mishkin[, 1]) > inflation_changes <- diff(inflation) > ts.plot(inflation) ; ts.plot(inflation_changes) inflation inflation_changes Time

10 MA Processes: Changes in Inflation Rate - II Inflation_changes: changes in one-month US inflation rate Plot the series and its sample ACF: > ts.plot(inflation_changes) > acf(inflation_changes, lag.max = 24) inflation_changes ACF Time Lag

11 MA Processes: Changes in Inflation Rate - III t WhiteNoise(0, 2 ) > MA_inflation_changes <- arima(inflation_changes, order = c(0, 0, 1)) > print(ma_inflation_changes) Coefficients: ma1 intercept s.e sigma^2 estimated as ma1, intercept, sigma^2

12 MA Processes: Fitted Values - I MA fitted values: Residuals= ˆ t = Y t Ŷ t

13 MA Processes: Fitted Values - II > ts.plot(inflation_changes) > MA_inflation_changes_fitted <- inflation_changes - residuals(ma_inflation_changes) > points(ma_inflation_changes_fitted, type = "l", col = "red", lty = 2) inflation_changes Time

14 Forecasting 1-step ahead forecasts: > predict(ma_inflation_changes) $pred Jan $se Jan h-step ahead forecasts: > predict(ma_inflation_changes, n.ahead = 6) $pred Jan Feb Mar Apr May Jun $se Jan Feb Mar Apr May Jun

15 INTRODUCTION TO TIME SERIES ANALYSIS Let s practice!

16 INTRODUCTION TO TIME SERIES ANALYSIS Compare the AR and MA models

17 MA and AR processes MA model: AR model: (Today Mean)=Slope (Yesterday Mean)+Noise Where: Y t µ = (Y t 1 µ)+ t t WhiteNoise(0, 2 )

18 MA and AR Processes: Autocorrelations MA: θ = 0.75 AR: φ = 0.48 ρ(h) ρ(h) h h MA: θ = 0.75 AR: φ = 0.48 ρ(h) ρ(h) h h

19 MA and AR Processes: Simulations MA: θ = 0.75 AR: φ = 0.48 Y t Time (t) MA: θ = 0.75 Y t Y t Time (t) AR: φ = 0.48 Y t Time (t) Time (t)

20 MA and AR Processes: Fitted Values Changes in one-month US inflation rate Inflation Changes inflation_changes MA fit AR fit Year

21 MA and AR Processes: Forecasts Changes in one-month US inflation rate Inflation Changes inflation_changes MA forecasts AR forecasts Year

22 Forecasting Akaike Information Criterion > MA_inflation_changes <- arima(inflation_changes, order = c(0,0,1)) ma1 intercept s.e sigma^2 estimated as 8.882: log likelihood = , aic = > AIC(MA_inflation_changes) > BIC(MA_inflation_changes) Bayesian Information Criterion > AR_inflation_changes <- arima(inflation_changes, order = c(1,0,0)) ar1 intercept s.e sigma^2 estimated as 10.37: log likelihood = , aic = > AIC(AR_inflation_changes) > BIC(AR_inflation_changes)

23 INTRODUCTION TO TIME SERIES ANALYSIS Let s practice!

24 INTRODUCTION TO TIME SERIES ANALYSIS Congratulations!

25 What you ve learned Manipulating ts objects, including log() and diff() Time series models: white noise, random walk, autoregression, simple moving average Time series simulation (arima.sim), fitting (arima), and forecasting (predict) Inflation Changes inflation_changes MA forecasts AR forecasts Year

26 INTRODUCTION TO TIME SERIES ANALYSIS Thank you!

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