ST4064 Practicals and case studies. Practical 1 Visualisation of univariate time series: stock market data

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1 ST4064 Practicals and case studies Practical 1 Visualisation of univariate time series: stock market data Extract and plot the EU Stock price data available in the R dataset EuStockMarkets (run help(eustockmarkets) for more information). 1. Plot the raw data for each. 2. Calculate and plot the increments (first- difference) of the series do they appear stationary? 3. Compute the log- series and their increments and plot them. 4. Of the above three representations of the 4 EuStockMarkets time series, which appear to be stationary? Why? 5. Plot ACFs for the log- difference series (use acf()) and comment on the pattern you obtain. 6. Using z=rnorm(1860); acf(z) as a reference, do the plots in (5) look like white noise? Create a 5- page document with plots (4 to a page with y- axes labeled). Annotate to answer the questions. 1

2 Practical 2 Simulating autoregressive time series Generate the following three AR time series for t = 1, 2,..., 1000 in R, where the realisations {e(t)} are i.i.d. Normal(0,1). For each series, plot (on one page) the data and the corresponding autocorrelation and partial autocorrelation functions. 1. x(t) =.5*x(t- 1) -.2*x(t- 2) +e(t) 2. x(t) =.8*x(t- 1) +e(t) 3. x(t) = 1.01*x(t- 1) +e(t) Create a document with answers to the questions. Annotate as you see fit. 2

3 Practical 3 Simulating ARMA models Consider the ARMA(2,2) model where x[t] <- a1* x[t- 1] + a2* x[t- 2]+e[t] + b1*e[t- 1] + b2*e[t- 2] a1<-.8 ; a2<- -.6 b1<-.6 ; b2< Generate a realization of length m=1000 and plot the sample, its acf and pacf. 2. Do the data look stationary (visually check the mean and variance)? : Create a document with answers to the questions. Annotate as you see fit. Case study: Lake Huron data Plot the LakeHuron Height Data (data(lakehuron)) together with its ACF and PACF. Suggest an ARMA model for this time series. Take the previous document for practical 3, and create a new section dedicated to this case study. Start by presenting the dataset and objective of the study. Then describe briefly the methodology you applied, providing plots and/or numerical values as appropriate, and your conclusion. Annotate as you see fit. 3

4 Practical 4 Transformation of time series Consider the following four series: (i) (ii) (iii) (iv) AirPassengers [Monthly Airline Passenger Numbers ] UKgas [Quarterly UK gas consumption from 1960Q1 to 1986Q4, in millions of therms] Nile [Measurements of the annual flow of the river Nile at Ashwan ] JohnsonJohnson [Quarterly earnings (dollars) per Johnson & Johnson share ] 1. Use ts.plot to plot each series on the same page. 2. Log- transform the series and plot again. 3. Apply differencing with diff(x,lag=k) using successively k=1, 4, and 12 to remove the trends and plot the differenced series (1- page plot). 4. Plot the ACFs and PACFs of the series in (3). 5. Use (4) to suggest possible ARMA models for each of the transformed series. Create a document with answers to the questions. Annotate as you see fit. 4

5 Practical 5 Fitting ARIMA models Consider the four series of Practical 4. Use the arima function in R to fit time series models to all four datasets. For each series: 1. Write down the model selected for the transformed data X as ARMA(p,q). 2. Give the difference equation for the mean adjusted series x = X- mean(x): 3. Use x[t] - phi[1]*x[t- 1] phi[p]*x[t- p] = e[t]+theta[1]*e[t- 1]+... +theta[q]*e[t- q] fit <- arima(x,order=c(p,0,q)) print(fit) to get the coefficients +- SE for your model parameters in (1). 4. Consider the residuals from your fitted model (r <- fit$residuals). Get plots for the residual time series, their ACF and PACF, and a Normal qq- plot as par(mfrow=c(2,2)) ts.plot(r) acf(r) pacf(r) qqnorm(r) abline(a=mean(r), b=sqrt(var(r))) Create a document with a 1- page description of the models identified, and a detailed summary of residual diagnostics. Annotate to answer the questions. 5

6 Practical 6 Detrending and smoothing Consider the log- transformed series of the AirPassengers dataset. 1. Detrend the series by linear regression. 2. Smooth the series first, using function filter with weights w <- c(-3,-6,-5,3,21,46,67,74,67,46,21,3,-5,-6,-3)/320 and then detrend by linear regression. 3. Use the seasonal = list(order=c(0,1,1)) command to include an MA term in the AirPassengers data. Redo the diagnostics for this model. 4. Check your fit by analysing the residuals as a stationary time series. 5. Could we compare the following two fits? > f1 = arima(x, order=c(0,1,1), seasonal=list(order=c(0,1,1))) > f2 = arima(x, order=c(13,0,13)) Plot over 3 pages: - the original time series, the logged series and the detrended series from question (1); - the logged series and corresponding smoothed series (together in one, using ts.plot), the detrended series and the detrended smoothed series from question (2); - the plot, ACF and PACF of the reiduals obtained from the fit in (3). Case study: forecasting UKgas, Nile and JohnsonJohnson data Consider again the series UKgas, Nile and JohnsonJohnson. Use the predict.arima function to obtain a 5- step ahead forecasts for the models identified for each time series as in question (3) above. Take the previous document for practical 3, and create a new section dedicated to this case study. Start by presenting the dataset and objective of the study. Then describe briefly the methodology you applied, providing plots and/or numerical values as appropriate, and your conclusion. Annotate as you see fit. 6

7 Practical 7 Fitting, performing diagnostic checks and forecasting 1. Simulate n=100 measurements of an AR(2) model with coefficients ar=c(0.8897, ) using arima.sim. Then: i) Generate two fits: an ARIMA(1,0,0) fit f1 and an ARIMA(2,0,0) fit f2 ii) Use tsdiag to observe the p- values of the Ljung- Box version of the Portmanteau test iii) Obtain 10- step ahead predictions p1 and p2 using each fit, and plot forecasts using L = p1$pred - 2*p1$se U = p1$pred + 2*p1$se miny = min(y,l) maxy = max(y,u) ts.plot(y,p1$pred,col=1:2,ylim=c(miny,maxy)) lines(u, col="blue", lty="dashed") lines(l, col="blue", lty="dashed") 2. Using each of the following two fits: f1 <- arima(log(ukgas), c(2,1,1), seasonal = list(order=c(0,1,1),period=4)) f2 <- arima(log(ukgas), c(2,1,1)) i) Get an 8- step ahead forecast and plot as in Question 1- iii ii) Check the residuals using the Portmanteau test iii) Check the residuals using Box.test with argument type="ljung- Box" 3. Seasonal decomposition (again): apply stl to log(airpassengers). Plot its output. Create a a 5- page document with: - tsdiag plot for Question 1- ii - plot of the two forecasts in Question 1- iii - plot of the two forecasts in Question 2- i - plot of residual diagnostic for f1 in Question 2- ii - plot of output of stl in Question 3 Annotate to answer the questions. 7

8 Practical 8 Multivariate time series modelling 1. Consider the USArrests data in R. (i) (ii) (iii) (iv) Plot the component variables against each other. Compute the correlation matrix of the data. Use function eigen to decompose the correlation matrix. Compare the result with prcomp(usarrests, scale = TRUE). 2. Consider the EU Stockprice data [library(ts); data(eustockmarkets)]. (i) (ii) (iii) Find suitable ARIMA models for each component. Present the model and the residual diagnostics. Use pairs() to plot each of the series against each other. Apply a principal component transformation out= prcomp(eustockmarkets,scale=true) Let z=out$x be the 4- dimensional dataset like EustockMarkets but which components are uncorrelated. Use pairs(z) to compare the variables. (iv) Starting with z[,4], find suitable ARIMA models for each of the components of z. Present the model and the residual diagnostics Create a document to answer the questions. 8

9 Practical 9 Simulating ARCH processes Simulate an ARCH(1) model x(t) using successively the following parameters: (i) mu=1 ; a0= 1 ; a1=.125 (ii) mu=1 ; a0= 1 ; a1=.5 (iii) mu=1 ; a0= 1 ; a1=2 In each case display the series for N=250, its ACF and PACF. Comment on stationarity. Create a 3- page document with the output for each case on each page. 9

10 Final instructions Once you have completed all practicals, combine all relevant documents, name the master file using your name and student number, and send it by to Before the end of term. 10