Computational Finance II: Time Series K.Ensor

What is a time series? Anything observed sequentially (by time?) Returns, volatility, interest rates, exchange rates, bond yields, … Hourly temperature, variations of the thickness of a wire as a function of length

What is different? The observations are not independent. There is correlation from observation to observation. Consider the log of the J&J series. Is there correlation in the observations over time?

What are our objectives? Understanding / Modeling Estimating summary measures (e.g. mean) Prediction / Forecasting If correlation is present between the observations then our typical approaches are not correct (assume iid samples).

Autocorrelation? How would you determine or show correlation over time?

Autocorrelation Function In theory… How to estimate this quantity?

Sample ACF and PACF Sample ACF – sample estimate of the autocorrelation function. Substitute sample estimates of the covariance between X(t) and X(t+h). Note: We do not have “n” pairs but “n-h” pairs. Subsitute sample estimate of variance. Sample PACF – correlation between observations X(t) and X(t+h) after removing the linear relationship of all observations in that fall between X(t) and X(t+h).

Summary Plots

Detrending by taking first difference. Y(t)=X(t) – X(t-1) What happens to the trend? Suppose X(t)=a+bt+Z(t) Z(t) is a random variable.

Detrended J&J series: Autocorrelation?

Sumary Plots of Detrended J&J log earnings per share.

Removing Seasonal Trend – one way to proceed. Suppose Y(t)=g(t)+W(t) where g(t)=g(t-s) where s is our “season” for all t. W(t) is again a new random variable Form a new series U(t) by taking the “s” difference U(t)=Y(t)-Y(t-s) =g(t)-g(t-s) + W(t)-W(t-s) =W(t)-W(t-s) again a random variable

Summary of Transformed J&J Series

Summary of Transformations: X(t) = log (Q(t)) Y(t)=X(t)-X(t-1) = (1-B)X(t) U(t)= (1-B 4 )Y(t) U(t)=(1-B 4 ) (1-B)X(t)

What is the next step? U(t) is a time series process called a moving average of order 1 (or possibly a MA(1) plus a seasonal MA(1)) U(t)=(t-1) + (t) Proceed to estimate and then we can estimate summary information about the earnings per share as well as predict.

Forecast of J&J series

Why does the autocorrelation matter when making inferences? Consider estimation of the mean of a stationary series E[X(t)]= for all t If X(1),…,X(n) are iid what is the sampling distribution of the estimator for , namely the sample mean?

Why does the autocorrelation matter? What if X(t) has the following structure (autoregressive model of order 1 AR(1) ) X(t)- =  (X(t-1)- ) +  (t) Then Corr(X(t),X(t+h))=  |h| for all h And Var(X)= (1+  Var(X)/n

Comparing the samples size? Let m denote the number of iid obs. Let n denote the number of correlated obs. Setting the variances equal and solving for m as a function of n yields m=n(1-  Let n=100, then m=5 iid obs. If n=100 and  then the equivalent number of iid observations is For positive and negative  (correlation of lag 1) the equivalent sample sizes are 33 and 300.

Why? Why does the autocorrelation make such a big difference in our ability to estimate the mean? The same arguments for other mean functions of the process or other functions of the process we want to estimate.

Summary Times series is correlated data, sequentially observed. The autocorrelation is a measure of the this correlation over the time lag. This dependence structure along with proper assumptions allows us to forecast the future of the process. Correct inference requires incorporating knowledge of the dependence structure.