1 Forecasting House Starts

Transcription

1 1396, Time Series, Week 5, Fall In this handout, we will see the application example on chapter 5. We use the same example as illustrated in the textbook and fit the data with several models of interests. Data: Housing starts are usually considered to be seasonal. Using monthly data on U.S. housing starts, we will estimate the regressions using the period and the period for out-of-sample forecasting. Library in R: additionally, only need to add-on library(car) for Durbin- Watson. 1 Forecasting House Starts 1.1 Read data into R > library(car) ## for Durbin.Watson test and Cook Distance > hs <- read.table("hstarts.dat", header=true) > hstarts <- hs$hstarts[1:576] 1.2 Time Series Plot Time series plot using time scale, Time = 1, 2,..., T. The plot is generated by the R command: > plot(hstarts, xlab="time") hstarts Time Times series plot using original time scale. commands: The plot is generated by the R > ## Create a time series with months attached to the data, from > hstarts.ts <- ts(hstarts, frequency=12, start=c(1946,1), end=c(1993,12)) > plot(hstarts.ts)

2 1396, Time Series, Week 5, Fall hstarts.ts Time 1.3 Monthly Effects: Seasonal variations in which months? To assess effects of all 12 months, we first create 12 dummy variables as follows. > n <- length(sales) ## n=576 ## > Time <- c(1:n) > d1 <- rep(c(1, rep(0,11)),48) > d2 <- rep(c(0,1,rep(0,10)), 48) > d3 <- rep(c(0,0,1,rep(0,9)), 48) > d4 <- rep(c(0,0,0,1, rep(0,8)),48) > d5 <- rep(c(rep(0,4),1,rep(0,7)), 48) > d6 <- rep(c(rep(0,5), 1, rep(0,6)), 48) > d7 <- rep(c(rep(0,6),1,rep(0,5)),48) > d8 <- rep(c(rep(0,7), 1, rep(0,4)), 48) > d9 <- rep(c(rep(0,8), 1,0,0,0),48) > d10 <- rep(c(rep(0,9),1,0,0), 48) > d11 <- rep(c(rep(0,10),1, 0), 48) > d12 <- rep(c(rep(0,11),1), 48) Check out one dummy variable to see the idea. > d1 [1] [38] [75] [112] [149] [186] [223]

3 1396, Time Series, Week 5, Fall [260] [297] [334] [371] [408] [445] [482] [519] [556] The model assessing the effects of all months is > hs.fit <- lm(hstarts~-1+d1+d2+d3+d4+d5+d6+d7+d8+d9+d10+d11+d12) > summary(hs.fit) Call: lm(formula = hstarts ~ -1 + d1 + d2 + d3 + d4 + d5 + d6 + d7 + d8 + d9 + d10 + d11 + d12) Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr(> t ) d <2e-16 *** d <2e-16 *** d <2e-16 *** d <2e-16 *** d <2e-16 *** d <2e-16 *** d <2e-16 *** d <2e-16 *** d <2e-16 *** d <2e-16 *** d <2e-16 *** d <2e-16 *** --- Signif. codes: 0 *** ** 0.01 * Residual standard error: on 564 degrees of freedom Multiple R-Squared: , Adjusted R-squared: F-statistic: on 12 and 564 DF, p-value: < 2.2e-16 Check the residuals for (1) Normal assumptions (2) Constant Variance (3) Independence

4 1396, Time Series, Week 5, Fall Normal Q Q Plot Sample Quantiles hs.fit$residuals Theoretical Quantiles hs.fit$fitted.values hs.fit$residuals e(t 1) Time e_(t) Tests for serial correlation > durbin.watson(hs.fit, method="normal", alternative="two.sided") lag Autocorrelation D-W Statistic p-value Alternative hypothesis: rho!= 0 > durbin.watson(hs.fit, method="normal", alternative="positive") lag Autocorrelation D-W Statistic p-value Alternative hypothesis: rho > 0 Now, we attempt to find AIC and SIC (we use log version shown on page 101 of the textbook) > hs.aic <- log(sum(hs.fit$residuals^2)/n)+2*12/n > hs.sic <- log(sum(hs.fit$residuals^2)/n)+12*log(n)/n > hs.aic [1] > hs.sic [1]

5 1396, Time Series, Week 5, Fall Notice that k in AIC and SIC now is 12, because there are 12 dummy variables in the model. The following shows the forecasting values of the next 12 months and the prediction intervals: > hs.new <- data.frame(d1=c(1, rep(0,11)), d2=c(0,1,rep(0,10)), + d3=c(0,0,1,rep(0,9)), d4=c(0,0,0,1,rep(0,8)), + d5=c(0,0,0,0,1,rep(0,7)), d6=c(rep(0,5),1,rep(0,6)), + d7=c(rep(0,6),1,rep(0,5)), d8=c(rep(0,7),1,rep(0,4)), + d9=c(rep(0,8),1,0,0,0), d10=c(rep(0,9),1,0,0), + d11=c(rep(0,10),1,0), d12=c(rep(0,11),1)) > > hs.pred.plim <- predict(hs.fit, hs.new, interval="prediction") > hs.pred.clim <- predict(hs.fit, hs.new, interval="confidence") > hs.pred.plim fit lwr upr > hs.pred.clim fit lwr upr The plot with forecast and prediction interval: > plot(time,hstarts,type="l",xlim=c(1,590),ylim=c(0,250)) > lines((n+1):(n+12), hs.pred.plim[,1],lty=2)

6 1396, Time Series, Week 5, Fall > lines((n+1):(n+12), hs.pred.plim[,2],lty=2) > lines((n+1):(n+12), hs.pred.plim[,3],lty=2) hstarts Time 1.4 Single Month Effect Suppose we are interested in the effect of June only. We will need two dummy variables. First, we can create a dummy variable for June: { 1, the month is June D 1 = 0, the month is not June The other dummy variable D 2 shows the Non-June effect. That is, D 2 = 1 D 1. The fitted model is > d1.june <- rep(c(rep(0,5),1,rep(0,6)),48) > d1.notjune <- rep(1,576)-d1 > hs.june.fit <- lm(hstarts~-1+d1.june+d1.notjune) > summary(hs.june.fit) Call: lm(formula = hstarts ~ -1 + d1.june + d1.notjune) Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr(> t ) d1.june <2e-16 ***

7 1396, Time Series, Week 5, Fall d1.notjune <2e-16 *** --- Signif. codes: 0 *** ** 0.01 * Residual standard error: on 574 degrees of freedom Multiple R-Squared: , Adjusted R-squared: F-statistic: 3678 on 2 and 574 DF, p-value: < 2.2e-16 Check for the residual plots Normal Q Q Plot Theoretical Quantiles Sample Quantiles hs.june.fit$fitted.values hs.june.fit$residuals Time hs.june.fit$residuals e_(t) e(t 1) Tests for serial correlation: That is, is e t correlated to e t 1? > durbin.watson(hs.june.fit, method="normal", alternative="two.sided") lag Autocorrelation D-W Statistic p-value Alternative hypothesis: rho!= 0 > durbin.watson(hs.june.fit, method="normal", alternative="positive") lag Autocorrelation D-W Statistic p-value Alternative hypothesis: rho > 0

8 1396, Time Series, Week 5, Fall AIC and BIC values: > hs.june.aic <- log(sum(hs.june.fit$residuals^2)/n)+2*2/n > hs.june.sic <- log(sum(hs.june.fit$residuals^2)/n)+2*log(n)/n > hs.june.aic [1] > hs.june.sic [1] Notice that the AIC and SIC are higher than the AIC, SIC from the model including all 12 dummy variables. 1.5 Adding Trend to the Model Now, in addition to the 12 dummy variables assessing the monthly effects, we are interested in whether the housing starts has relationship with Time. So we add a linear Time trend into the model. The fitted regression is > hs2.fit <- lm(hstarts~-1+d1+d2+d3+d4+d5+d6+d7+d8+d9+d10+d11+d12+time) > summary(hs2.fit) Call: lm(formula = hstarts ~ -1 + d1 + d2 + d3 + d4 + d5 + d6 + d7 + d8 + d9 + d10 + d11 + d12 + Time) Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr(> t ) d e e <2e-16 *** d e e <2e-16 *** d e e <2e-16 *** d e e <2e-16 *** d e e <2e-16 *** d e e <2e-16 *** d e e <2e-16 *** d e e <2e-16 *** d e e <2e-16 *** d e e <2e-16 *** d e e <2e-16 *** d e e <2e-16 *** Time 1.160e e

9 1396, Time Series, Week 5, Fall Signif. codes: 0 *** ** 0.01 * Residual standard error: on 563 degrees of freedom Multiple R-Squared: , Adjusted R-squared: F-statistic: on 13 and 563 DF, p-value: < 2.2e-16 Check for residuals plots Normal Q Q Plot Sample Quantiles hs2.fit$residuals Theoretical Quantiles hs2.fit$fitted.values hs2.fit$residuals e(t 1) Index e_(t) Normality test: > shapiro.test(hs2.fit$residuals) Shapiro-Wilk normality test data: hs2.fit$residuals W = , p-value = 4.047e-09 AIC and SIC:

10 1396, Time Series, Week 5, Fall > hs2.aic <- log(sum(hs2.fit$residuals^2)/n)+2*13/n > hs2.sic <- log(sum(hs2.fit$residuals^2)/n)+13*log(n)/n > hs2.aic [1] > hs2.sic [1] Conclusions It appears that there were higher housing starts in the period of April to June. Time trend is not shown to be significant. The AIC and SIC values are slightly lower when the Time trend is not included. The normal plot is not improved too much if adding the linear Time trend in the model. Exercise 1. It appears that there were higher housing starts in the period of April to June. Build a model to assess the effects of this period by doing the followings. (a) How to create dummy variables for effects of April, May and June? To assess the effects, there should be four dummy variables for putting into the model. Write out these four dummy variables. (b) Using the data from the period , fit the regression model on these four dummy variables. Present your regression and give the estimates of those effects from the regression. (c) What are the p-values for the t-tests shown in the output? Write down the hypothesis for each t-test and draw conclusions about the significance. (d) Do the residuals support your model assumptions? Check out the residual plot, including qqnorm plot, residuals-time plot and a plot for detecting serial correlation. Also carry out Durbin-Watson test on the residuals. (e) Find out AIC and SIC and compare them with the models shown in this handout. (f) Provide the prediction intervals and confidence intervals for the forecasts at April, May and June in 1994.