Financial time series (1)

Lecture Note: Analysis of Financial Time Series Spring 2015, Ruey S. Tsay Seasonal Time Series: TS with periodic patterns and useful in • predicting quarterly earnings • pricing weather-related derivatives • analysis of transactions data (high-frequency data), e.g., U-shaped pattern in intraday trading intensity, volatility, etc. Example 1. Monthly U.S. Housing Starts from January 1959 to February 2012. The data are in thousand units. See Figure 1 and compute the sample ACF of the series and its differenced data. Example 2. Quarterly earnings of Johnson & Johnson See the time plot, Figures 2 and 3, and sample ACFs Example 3. Quarterly earning per share of Coca Cola from 1983 to 2009. Multiplicative model: Consider the housing-starts series. Let yt be the monthly data. Denoting 1959 as year 0, we can write the time index as t = year + month, e.g, y1 = y0,1, y2 = y0,2, and y14 = y1,2, etc. The multiplicative model is based on the following consideration: 1 200 150 50 100 hst 1960 1970 1980 1990 2000 2010 Time Figure 1: Time plot of monthly U.S. housing starts: 1959.1-2012.2. Data obtained from US Bureau of the Census. 0 5 x 10 15 Quarterly earnings of JNJ: 1960−1980 1960 1965 1970 Time 1975 1980 Figure 2: Time plot of quarterly earnings of Johnson and Johnson: 1960-1980 2 2 y 1 0 1960 1965 1970 Time 1975 1980 Figure 3: Time plot of quarterly logged earnings of Johnson and Johnson: 1960-1980 0.0 0.2 0.4 y 0.6 0.8 1.0 EPS of Coca Cola: 1983−2009 1985 1990 1995 2000 2005 2010 Time Figure 4: Time plot of quarterly earnings per share of KO (Coca Cola) from 1983 to 2009. 3 Month Year Jan Feb Mar · · · Oct Nov Dec 1959 y0,1 y0,2 y0,3 · · · y0,10 y0,11 y0,12 1960 y1,1 y1,2 y1,3 · · · y1,10 y1,11 y1,12 1961 y2,1 y2,2 y2,3 · · · y2,10 y2,11 y2,12 1962 y3,1 y3,2 y3,3 · · · y3,10 y3,11 y3,12 ... ... ... ... ... ... ... The column dependence is the usual lag-1, lag-2, ... dependence. That is, monthly dependence. We call them the regular dependence. The row dependence is the year-to-year dependence. We call them the seasonal dependence. Multiplicative model says that the regular and seasonal dependence are orthogonal to each other. Airline model for quarterly series • Form: rt − rt−1 − rt−4 + rt−5 = at − θ1at−1 − θ4at−4 + θ1θ4at−5 or (1 − B)(1 − B 4)rt = (1 − θ1B)(1 − θ4B 4)at • Define the differenced series wt as wt = rt − rt−1 − rt−4 + rt−5 = (rt − rt−1) − (rt−4 − rt−5). It is called regular and seasonal differenced series. 4 3 2 1 0 −1 0 20 40 60 80 Figure 5: Forecast plot for the quarterly earnings of Johnson and Johnson. Data: 1960-1980, Forecasts: 1981-82. • ACF of wt has a nice symmetric structure (see the text), i.e. ρs−1 = ρs+1 = ρ1ρs. Also, ρℓ = 0 for ℓ > s + 1. • This model is widely applicable to many many seasonal time series. • Multiplicative model means that the regular and seasonal dependences are roughly orthogonal to each other. • Forecasts: exhibit the same pattern as the observed series. See Figure 5. • Exponential Smoothing method Example Detailed analysis of J&J earnings. R Demonstration: output edited. 5 > > > > > > > > > > > > > > x=ts(scan("q-earn-jnj.txt"),frequency=4,start=c(1960,1)) % create a time series object. plot(x) % Plot data with calendar time y=log(x) % Natural log transformation plot(y) % plot data c1=paste(c(1:4)) points(y,pch=c1) % put circles on data points. par(mfcol=c(2,1)) % two plots per page acf(y,lag.max=16) y1=as.vector(y) % Creates a sequence of data in R acf(y1,lag.max=16) dy1=diff(y1) % regular difference acf(dy1,lag.max=16) sdy1=diff(dy1,4) % seasonal difference acf(sdy1,lag.max=12) > m1=arima(y1,order=c(0,1,1),seasonal=list(order=c(0,1,1),period=4)) % Airline % model in R. > m1 Call: arima(x = y1, order = c(0, 1, 1), seasonal = list(order = c(0, 1, 1), period = 4)) Coefficients: ma1 -0.6809 s.e. 0.0982 sma1 -0.3146 0.1070 % The fitted model is (1-B^4)(1-B)R(t) = % (1-0.68B)(1-0.31B^4)a(t), var[a(t)] = 0.00793. sigma^2 estimated as 0.00793: log likelihood = 78.38, aic = -150.75 > par(mfcol=c(1,1)) % One plot per page > tsdiag(m1) % Model checking > f1=predict(m1,8) % prediction > names(f1) [1] "pred" "se" > f1 $pred % Point forecasts Time Series: Start = 85 End = 92 Frequency = 1 [1] 2.905343 2.823891 2.912148 2.581085 3.036450 2.954999 3.043255 2.712193 $se % standard errors of point forecasts Time Series: Start = 85 End = 92 Frequency = 1 [1] 0.08905414 0.09347895 0.09770358 0.10175295 0.13548765 0.14370550 6 [7] 0.15147817 0.15887102 # You can use ‘‘foreplot’’ to obtain plot of forecasts. For monthly data, the Airline model becomes (1 − B)(1 − B 12)rt = (1 − θ1B)(1 − θ12B 12)at. What is the pattern of ACF? Regression Models with Time Series Errors Question: Why don’t we use R-square in this course? • Has many applications • Impact of serial correlations in regression is often overlooked. It may introduce biases in estimates and in standard errors, resulting in unreliable t-ratios. • Detecting residual serial correlation: Use Q-stat instead of DWstatistic, which is not sufficient! • Joint estimation of all parameters is preferred. • Avoid the problem of spurious regression. R-square can be misleading!!! • Proper analysis: see the illustration below. Example. U.S. weekly interest rate data: 1-year and 3-year constant maturity rates. Data are shown in Figure 6. R Demonstration: output edited. 7 15 percent 10 5 1970 1980 year 1990 2000 Figure 6: Time plots of U.S. weekly interest rates: 1-year constant maturity rate (solid line) and 3-year rate (dashed line). > > > > > > da=read.table("w-gs1n36299.txt") % load the data r1=da[,1] % 1-year rate r3=da[,2] % 3-year rate plot(r1,type=’l’) % Plot the data lines(1:1967,r3,lty=2) plot(r1,r3) % scatter plot of the two series > m1=lm(r3~r1) % Fit a regression model with likelihood method. > summary(m1) Call: lm(formula = r3 ~ r1) Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 0.910687 0.032250 28.24 <2e-16 *** r1 0.923854 0.004389 210.51 <2e-16 *** --Signif. codes: 0 ’***’ 0.001 ’**’ 0.01 ’*’ 0.05 ’.’ 0.1 ’ ’ 1 Residual standard error: 0.538 on 1965 degrees of freedom Multiple R-Squared: 0.9575, Adjusted R-squared: 0.9575 F-statistic: 4.431e+04 on 1 and 1965 DF, p-value: < 2.2e-16 8 > > > > acf(m1$residuals) c3=diff(r3) c1=diff(r1) plot(c1,c3) > m2=lm(c3~c1) % Fit a regression with likelihood method. > summary(m2) Call: lm(formula = c3 ~ c1) Residuals: Min 1Q Median -0.3806040 -0.0333840 -0.0005428 3Q 0.0343681 Max 0.4741822 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 0.0002475 0.0015380 0.161 0.872 c1 0.7810590 0.0074651 104.628 <2e-16 *** --Residual standard error: 0.06819 on 1964 degrees of freedom Multiple R-Squared: 0.8479, Adjusted R-squared: 0.8478 F-statistic: 1.095e+04 on 1 and 1964 DF, p-value: < 2.2e-16 > acf(m2$residuals) > plot(m2$residuals,type=’l’) > m3=arima(c3,xreg=c1,order=c(0,0,1)) % Residuals follow an MA(1) model > m3 Call: arima(x = c3, order = c(0, 0, 1), xreg = c1) Coefficients: ma1 intercept 0.2115 0.0002 s.e. 0.0224 0.0018 c1 0.7824 0.0077 sigma^2 estimated as 0.004456: > acf(m3$residuals) > tsdiag(m3) % Fitted model is % c3 = 0.0002+0.782c1 + a(t)+0.212a(t-1) % with var[a(t)] = 0.00446. log likelihood = 2531.84, aic = -5055.69 > m4=arima(c3,xreg=c1,order=c(1,0,0)) % Residuals follow an AR(1) model. > m4 Call: arima(x = c3, order = c(1, 0, 0), xreg = c1) Coefficients: ar1 intercept c1 % Fitted model is 9 s.e. 0.1922 0.0221 0.0003 0.0019 0.7829 0.0077 sigma^2 estimated as 0.004474: % c3 = 0.0003 + 0.783c1 + a(t), % a(t) = 0.192a(t-1)+e(t). log likelihood = 2527.86, aic = -5047.72 Remark: Parameterization in R. With additional explanatory variable X in ARIMA model, R use the model Wt = φ1Wt−1 + · · · + φpWt−p + at + θ1at−1 + · · · + θq at−q , where Wt = Yt − β0 − β1Xt. This is the proper way to handle regression model with time series errors, because Wt−1 is not subject to the effect of Xt−1. It is different from the model Yt = β0∗ + β1∗Xt + φ1Yt−1 + · · · + φpYt−p + at + θ1at−1 + · · · + θq at−q , for which the Yt−1 contains the effect of Xt−1. Long-memory models • Meaning? ACF decays to zero very slowly! • Example: ACF of squared or absolute log returns ACFs are small, but decay very slowly. • How to model long memory? Use “fractional” difference: namely, (1 − B)drt, where −0.5 < d < 0.5. • Importance? In theory, Yes. In practice, yet to be determined. • In R, the package rugarch may be used to estimate the fractionally integrated ARMA models. The package can also be used for GARCH modeling. 10 Summary of the chapter • Sample ACF ⇒ MA order • Sample PACF ⇒ AR order • Some packages have “automatic” procedure to select a simple model for “conditional mean” of a FTS, e.g., R uses “ar” for AR models. • Check a fitted model before forecasting, e.g. residual ACF and hetroscedasticity (chapter 3) • Interpretation of a model, e.g. constant term & For an AR(1) with coefficient φ1, the speed of mean reverting as measured by half-life is k= ln(0.5) . ln(|φ1|) For an MA(q) model, forecasts revert to the mean in q + 1 steps. • Make proper use of regression models with time series errors, e.g. regression with AR(1) residuals Perform a joint estimation instead of using any two-step procedure, e.g. Cochrane-Orcutt (1949). • Basic properties of a random-walk model • Multiplicative seasonal models, especially the so-called airline model. 11