Econ 526 Precipiation

1.

#annual data
Annual <- scan("http://www.calpoly.edu/~gbdurham/econ522/precip/annual_precip.csv")
Annual <- ts(Annual)

#monthly data
monthly <- scan("http://www.calpoly.edu/~gbdurham/econ522/precip/monthly_precip.csv")
monthly <- ts(monthly, frequency = 12, start = c(0,6))

2.

plot.ts(Annual)

3.

plot.ts(Annual)
s = ksmooth(1:145,Annual,bandwidth = 5)
lines(smooth(s$y),col = 3)

4.

res = Annual - s$y
par(mfrow=c(2,1))
plot(res)
hist(res)

par(mfrow=c(1,1))
acf(res)

#The residuals seem normal

4.

lamda <- 0.5
z = ((Annual^lamda)-1)/lamda
zSmooth<- ksmooth(1:145,z,bandwidth = 5)
Z_Res <- z - zSmooth$y
plot(Z_Res)

hist(Z_Res)

#Residuals at Lamda = 0.5 seem to be normal

5.

Monthly <- ts(monthly, frequency = 12, start = c(0,6))
monthplot(Monthly)

MonthSmooth<- ksmooth(1:145,Monthly,bandwidth = 5)
Monthly_Res <- Monthly - MonthSmooth$y
plot(Monthly_Res)

hist(Monthly_Res)

#does not seem to appear normal before transformation


lamda <- 0.5
z = ((Monthly^lamda)-1)/lamda
zSmooth<- ksmooth(1:145,Monthly,bandwidth = 5)
Z_Res <- z - zSmooth$y
plot(Z_Res)

hist(Z_Res)

#Seems to be normal after Box Cox Transformation

6.

Monthlyfactor <- factor(cycle(z))
Dummy1<- model.matrix(z~ Monthlyfactor)
Dummy1<- Dummy1[,-1]

auto.arima(z, xreg = Dummy1)

## Series: z 
## ARIMA(1,0,0)(2,0,2)[12] with non-zero mean 
## 
## Coefficients:

## Warning in sqrt(diag(x$var.coef)): NaNs produced

##          ar1     sar1    sar2    sma1     sma2  intercept  Monthlyfactor2
##       0.0879  -0.2526  0.4896  0.2630  -0.4521     1.7626         -0.3676
## s.e.  0.0141      NaN     NaN  0.0087   0.0154     0.1144          0.1546
##       Monthlyfactor3  Monthlyfactor4  Monthlyfactor5  Monthlyfactor6
##              -1.6098         -2.8217         -3.4596         -3.6535
## s.e.          0.1612          0.1618          0.1618          0.1618
##       Monthlyfactor7  Monthlyfactor8  Monthlyfactor9  Monthlyfactor10
##              -3.6335         -3.2443         -2.3462          -1.3704
## s.e.          0.1618          0.1618          0.1618           0.1618
##       Monthlyfactor11  Monthlyfactor12
##               -0.1761           0.1197
## s.e.           0.1612           0.1546
## 
## sigma^2 estimated as 1.671:  log likelihood=-2915.77
## AIC=5867.54   AICc=5867.93   BIC=5965.84

ARTrend<- arima(z, order = c(1,0,0), include.mean = FALSE, xreg = Dummy1)
ARTrend2<- arima(z, order = c(2,0,1), include.mean = FALSE, xreg = Dummy1)

## Warning in arima(z, order = c(2, 0, 1), include.mean = FALSE, xreg =
## Dummy1): possible convergence problem: optim gave code = 1

#AIC on ARMA (2,1) had the Lowest AIC of 5881.75
tsdiag(ARTrend)

tsdiag(ARTrend2)