Class: Dean Foster. September 30, Read sections: Examples chapter (chapter 3) Question today: Do prices go up faster than they go down?

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1 Class: Dean Foster September 30, 2013 Administrivia Read sections: Examples chapter (chapter 3) Gas prices Question today: Do prices go up faster than they go down? Idea is that sellers watch spot price and increase directly. But, only slow competition drives the price down. So there should be an asymetry Time reversed Concept: This means the process is not time reversable But if we reverse a series: The normality stays the same the covariance structure stays the same the mean stays the same So for a linear time series, a reversed process looks identical 1

2 If you can tell which direction is which it isn t linear! reading the data I like longer names than Reuy does: > crude = read.table("chapter3/w-petroprice.txt", header = TRUE) > gas = read.table("chapter3/w-gasoline.txt") > log.gas <- log(gas[, 1]) > log.crude <- log(crude$us) > date <- as.date(paste(crude$mon, crude$day, crude$year), format = "% Note date magic! It is worth playing around with dates since they come up often in time series. Simple plots > plot(date, log.gas) log.gas

3 > plot(date, log.crude) log.crude Looking at changes > delta.gas <- diff(log.gas) > delta.crude <- diff(log.crude) And some more pictures > acf(delta.gas, lag = 20) 3

4 Series delta.gas ACF Lag > pacf(delta.gas, lag = 20) 4

5 Series delta.gas Partial ACF Lag Model selection This is what you did for homework, so I m not going to cover it in detail. See the book for details. > m1 = ar(delta.gas, method = "mle") > m1$aic > m1$order [1] 5 5

6 Regression on crude > arima(delta.gas, order = c(6, 0, 0), include.mean = F, xreg = delta. Call: arima(x = delta.gas, order = c(6, 0, 0), xreg = delta.crude, include.me Coefficients: ar1 ar2 ar3 ar4 ar5 ar6 delta.crude s.e sigma^2 estimated as : log likelihood = , aic = my version of LAG > LAG <- function(x, lag = 1) { + n <- length(x) + return(c(rep(na, lag), x[1:(n - lag)])) + } Looking for non-linearities Nothing in the relationship with crude: > summary(lm(delta.gas ~ I(LAG(delta.gas)) + delta.crude + I(delta.cru + I(LAG(delta.crude)) + I(LAG(delta.crude)^2))) Call: lm(formula = delta.gas ~ I(LAG(delta.gas)) + delta.crude + I(delta.crud I(LAG(delta.crude)) + I(LAG(delta.crude)^2)) Residuals: Min 1Q Median 3Q Max 6

7 Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) I(LAG(delta.gas)) <2e-16 *** delta.crude <2e-16 *** I(delta.crude^2) I(LAG(delta.crude)) I(LAG(delta.crude)^2) Signif. codes: 0 '***' '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 Residual standard error: on 709 degrees of freedom (1 observation deleted due to missingness) Multiple R-squared: , Adjusted R-squared: F-statistic: on 5 and 709 DF, p-value: < 2.2e-16 How about in the delta.gas itself? > summary(lm(delta.gas ~ I(LAG(delta.gas)) + I(LAG(delta.gas)^2) + + I(LAG(delta.gas, 2)) + I(LAG(delta.gas, 3)) + I(LAG(delta.gas, + 4)) + I(LAG(delta.gas, 5)) + I(LAG(delta.crude)))) Call: lm(formula = delta.gas ~ I(LAG(delta.gas)) + I(LAG(delta.gas)^2) + I(LAG(delta.gas, 2)) + I(LAG(delta.gas, 3)) + I(LAG(delta.gas, 4)) + I(LAG(delta.gas, 5)) + I(LAG(delta.crude))) Residuals: Min 1Q Median 3Q Max Coefficients: 7

8 Estimate Std. Error t value Pr(> t ) (Intercept) *** I(LAG(delta.gas)) < 2e-16 *** I(LAG(delta.gas)^2) < 2e-16 *** I(LAG(delta.gas, 2)) I(LAG(delta.gas, 3)) ** I(LAG(delta.gas, 4)) I(LAG(delta.gas, 5)) * I(LAG(delta.crude)) Signif. codes: 0 '***' '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 Residual standard error: on 703 degrees of freedom (5 observations deleted due to missingness) Multiple R-squared: , Adjusted R-squared: F-statistic: on 7 and 703 DF, p-value: < 2.2e-16 > cubic <- lm(delta.gas ~ I(LAG(delta.gas)) + I(LAG(delta.gas)^2) + + I(LAG(delta.gas)^3) + I(LAG(delta.gas)^4) + I(LAG(delta.gas, + 2)) + I(LAG(delta.gas, 3)) + I(LAG(delta.gas, 4)) + I(LAG(delta. + 5)) + I(LAG(delta.crude))) > summary(cubic) Call: lm(formula = delta.gas ~ I(LAG(delta.gas)) + I(LAG(delta.gas)^2) + I(LAG(delta.gas)^3) + I(LAG(delta.gas)^4) + I(LAG(delta.gas, 2)) + I(LAG(delta.gas, 3)) + I(LAG(delta.gas, 4)) + I(LAG(delta.gas 5)) + I(LAG(delta.crude))) Residuals: Min 1Q Median 3Q Max

9 Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) 2.794e e *** I(LAG(delta.gas)) 6.709e e < 2e-16 *** I(LAG(delta.gas)^2) e e e-06 *** I(LAG(delta.gas)^3) e e * I(LAG(delta.gas)^4) 1.295e e I(LAG(delta.gas, 2)) 4.026e e I(LAG(delta.gas, 3)) 1.214e e ** I(LAG(delta.gas, 4)) e e I(LAG(delta.gas, 5)) e e * I(LAG(delta.crude)) e e Signif. codes: 0 '***' '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 > polynomial <- cbind(rep(1, length(delta.gas)), delta.gas, delta.gas^ Residual standard error: on 701 degrees of freedom (5 observations deleted due to missingness) Multiple R-squared: 0.429, Adjusted R-squared: F-statistic: on 9 and 701 DF, p-value: < 2.2e-16 > coef <- cubic$coefficients[1:5] > coef (Intercept) I(LAG(delta.gas)) I(LAG(delta.gas)^2) I(LAG(delta I(LAG(delta.gas)^4) delta.gas^3, delta.gas^4) %*% coef > plot(delta.gas, polynomial) > lines(delta.gas, delta.gas, col = "red") 9

10 delta.gas polynomial Different shocks? Our final test for an interpretable model of different shocks will consider the idea that positive shocks in crude should lead to different structure than negative shocks. > summary(lm(delta.gas ~ I(LAG(delta.gas, 1)) + I(LAG(delta.gas, + 2)) + I(LAG(delta.gas, 3)) + I(LAG(delta.gas, 1) * (LAG(delta.cr + 0)) + I(LAG(delta.gas, 2) * (LAG(delta.crude) > 0)) + I(LAG(delt + 3) * (LAG(delta.crude) > 0)))) Call: lm(formula = delta.gas ~ I(LAG(delta.gas, 1)) + I(LAG(delta.gas, 2)) + I(LAG(delta.gas, 3)) + I(LAG(delta.gas, 1) * (LAG(delta.crude 0)) + I(LAG(delta.gas, 2) * (LAG(delta.crude) > 0)) + I(LAG(delta.g 3) * (LAG(delta.crude) > 0))) 10

11 Residuals: Min 1Q Median 3Q Max Estimate Std. Error t v Coefficients: (Intercept) I(LAG(delta.gas, 1)) I(LAG(delta.gas, 2)) I(LAG(delta.gas, 3)) I(LAG(delta.gas, 1) * (LAG(delta.crude) > 0)) I(LAG(delta.gas, 2) * (LAG(delta.crude) > 0)) I(LAG(delta.gas, 3) * (LAG(delta.crude) > 0)) Pr(> t ) (Intercept) * I(LAG(delta.gas, 1)) < 2e-16 *** I(LAG(delta.gas, 2)) I(LAG(delta.gas, 3)) * I(LAG(delta.gas, 1) * (LAG(delta.crude) > 0)) *** I(LAG(delta.gas, 2) * (LAG(delta.crude) > 0)) I(LAG(delta.gas, 3) * (LAG(delta.crude) > 0)) Signif. codes: 0 '***' '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 Residual standard error: on 706 degrees of freedom (3 observations deleted due to missingness) Multiple R-squared: , Adjusted R-squared: F-statistic: on 6 and 706 DF, p-value: < 2.2e-16 How about just the AR(1) term model > summary(lm(delta.gas ~ I(LAG(delta.gas, 1)) + I(LAG(delta.gas, + 1)^2) + I(LAG(delta.gas, 1)^3) + I(LAG(delta.gas, 1) * (LAG(delt 11

12 + 0)) + I(LAG(delta.gas, 1)^2 * (LAG(delta.crude) > 0)) + I(LAG(de + 1)^3 * (LAG(delta.crude) > 0)))) Call: lm(formula = delta.gas ~ I(LAG(delta.gas, 1)) + I(LAG(delta.gas, 1)^2) + I(LAG(delta.gas, 1)^3) + I(LAG(delta.gas, 1) * (LAG(delta.c 0)) + I(LAG(delta.gas, 1)^2 * (LAG(delta.crude) > 0)) + I(LAG(delta 1)^3 * (LAG(delta.crude) > 0))) Residuals: Min 1Q Median 3Q Max Estimate Std. Error t Coefficients: (Intercept) 2.792e e-04 I(LAG(delta.gas, 1)) 6.845e e-02 I(LAG(delta.gas, 1)^2) e e+00 I(LAG(delta.gas, 1)^3) e e+01 I(LAG(delta.gas, 1) * (LAG(delta.crude) > 0)) e e-01 I(LAG(delta.gas, 1)^2 * (LAG(delta.crude) > 0)) 1.900e e+00 I(LAG(delta.gas, 1)^3 * (LAG(delta.crude) > 0)) 4.362e e+01 Pr(> t ) (Intercept) *** I(LAG(delta.gas, 1)) < 2e-16 *** I(LAG(delta.gas, 1)^2) 2.68e-06 *** I(LAG(delta.gas, 1)^3) ** I(LAG(delta.gas, 1) * (LAG(delta.crude) > 0)) I(LAG(delta.gas, 1)^2 * (LAG(delta.crude) > 0)) I(LAG(delta.gas, 1)^3 * (LAG(delta.crude) > 0)) Signif. codes: 0 '***' '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 12

13 Residual standard error: on 708 degrees of freedom (1 observation deleted due to missingness) Multiple R-squared: , Adjusted R-squared: F-statistic: on 6 and 708 DF, p-value: < 2.2e-16 > summary(lm(delta.gas ~ I(LAG(delta.gas, 1)) + I(LAG(delta.gas, + 1)^2) + I(LAG(delta.gas, 1)^3)))$r.squared [1]