Unemployment Time Series

Displaying the data

unemployment <- read.table ("unemployment.dat", header=TRUE)
ggplot(unemployment, aes(year, unemployed.pct)) + geom_line() + ylab("unemployed (%)")

Let’s try to fit a linear model:

unemployment$year1975 <- unemployment$year-1975
fit <- lm(unemployed.pct~year1975, data=unemployment)
summary(fit)

## 
## Call:
## lm(formula = unemployed.pct ~ year1975, data = unemployment)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -2.3325 -0.8997 -0.0546  0.7667  3.9057 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  5.58505    0.18727  29.823  < 2e-16 ***
## year1975     0.02990    0.01118   2.674  0.00981 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.426 on 56 degrees of freedom
## Multiple R-squared:  0.1132, Adjusted R-squared:  0.09739 
## F-statistic:  7.15 on 1 and 56 DF,  p-value: 0.009806

There is a pretty small upward trend. Let’s plot the residuals.

qplot(unemployment$year, fit$residuals)

It’s pretty clear that the residuals are not independent – large residuals tend to occur together.

Fitting the AR(1) model

Let’s fit the AR(1) model:

y <- unemployment$unemployed.pct
n <- length(y)
y.lag <- c(NA, y[1:(n-1)])
lm.lag <- lm(y~y.lag)

Do the residuals look normal?

qplot(unemployment$year[2:length(y)], lm.lag$residuals)

Kind of, and it seems like we got rid of most of the “parabola” pattern in the previous plot.

One way to see if the model is appropriate is “fake-data simulation”: simulate several datasets from the model using the fitted parameters.

b.hat <- coef (lm.lag)        # vector of 2 regression coefs
s.hat <- sigma.hat (lm.lag)   # residual sd


simulate_series <- function(){
  n_years <- 58
  y1 <- 3.9
  y.rep <- rep(0, n_years)
  y.rep[1] <- y1
  for (t in 2:n_years){
    prediction <- c (1, y.rep[t-1]) %*% b.hat
    y.rep[t] <- rnorm (1, prediction, s.hat)
  }
  ggdata <- data.frame(year=seq(1947, 2004), unemployment.pct=y.rep)
  return(y.rep)
}

n.sims <- 5
y.rep <- replicate(n.sims, simulate_series())

Let’s create a bunch of plots of simulations:

ggdata <- data.frame(year=seq(1947, 2004), unemployed.pct=y.rep[,1])
ggplot(ggdata, aes(year, unemployed.pct)) + geom_line() + ylab("unemployed (%)")

ggdata <- data.frame(year=seq(1947, 2004), unemployed.pct=y.rep[,2])
ggplot(ggdata, aes(year, unemployed.pct)) + geom_line() + ylab("unemployed (%)")

ggdata <- data.frame(year=seq(1947, 2004), unemployed.pct=y.rep[,3])
ggplot(ggdata, aes(year, unemployed.pct)) + geom_line() + ylab("unemployed (%)")

ggdata <- data.frame(year=seq(1947, 2004), unemployed.pct=y.rep[,4])
ggplot(ggdata, aes(year, unemployed.pct)) + geom_line() + ylab("unemployed (%)")