L7: ARIMA1 Lecture 7: ARIMA Model Process The following topics will be covered: Properties of Stock Returns AR model MA model ARMA Non-Stationary Process Seasonal Models Regression Models with Time Series Errors

L7: ARIMA2 Time Series Plot data msi; set crsp.msi; mret=vwretd; mno=(year(date)-1930)*12+month(date); keep mno mret; proc gplot data=msi; symbol v=none i=join l=1; plot mret*mno; run;

L7: ARIMA3 Basics Return measures Empirical properties of returns (page 16– Table 1.2 and Figure 1.4) –Daily returns of market indexes and individual stocks tend to have high excess kurtoses –The standard deviation of monthly returns is greater than the standard deviation of daily returns –The difference between simple and log returns is not substantial.

L7: ARIMA4 Stationarity A time series {r t } is said to be strictly stationary if the joint distribution {r t 1, …, r t k } is identical to that of {r t 1+t, …, r t k+t } for all t, where k is an arbitrary positive integer. –See page 9 to 10 of RT for the definition of joint distribution A time series {r t } is said to be weakly stationary if both the mean of rt and the covariance between r t and r t-l are time-invariant, where is an arbitrary integer. That is (1) E(r t )=μ (2) Var(rt)= γ 0 (3) cov(r t, r t-l )=γ l

L7: ARIMA5 Correlation and Auto-correlations

L7: ARIMA6 Portmanteau Statistic

L7: ARIMA7 Alternative Time Series

L7: ARIMA8 Properties of AR Models Mean and Variance of AR(1) –See page 30 – derive them Autocorrelation Function an AR(1) model Mean, Variance, and ACF of AR(2) – page 29 Stationarity of AR(2) AR(p) model

L7: ARIMA9 AR(2)

L7: ARIMA10 Identifying AR Models

L7: ARIMA11 Estimation of AR(p)

L7: ARIMA12 Forecasting AR(p)

L7: ARIMA13 More on Forecasting AR(p) 2) 2-step Ahead Forecast See page 40 – the variance of the forecast error is Var[e h (2)]= Model Checking: If the model is adequate, then the residual series should behave as a white noise. The ACF and the Portmanteau test of the residual {at} can be used to check the closeness of at to a white noise. proc arima data=msi; identify var=mret nlag=12; estimate p=1 method=ml; forecast lead=1; run;

L7: ARIMA14 Moving-Average (MA) Models

L7: ARIMA15 ARMA Models

L7: ARIMA16 Random Walk and ARIMA Random Walk without a Drift: p t =p t-1 +a t Random Walk with a drift: p t =μ+p t-1 +a t ARIMA – page 60

L7: ARIMA17 Seasonal Models Seasonality of quarterly earnings Seasonal adjustments: y t -y t-s =(1-B s )y t Multiplicative Seasonal Models

L7: ARIMA18 Regressions with Time Series Errors

L7: ARIMA19 Exercises (1) Ch 2, 6 (2) Get quarterly earnings per share (data11) from Quarterly Computstat file. Do the following: (a) Compute average EPS across all firms (b) Plot average quarterly EPS from 1996 through 2004 (c) Plot ACF and PACF for quarterly EPS (d) Take seasonal difference and check ACF and PACF again