1.10

#n=500
#x=numeric(n)
#w = rnorm(n)
#s = rep( c(.4, 0.8, -0.8, -.4 ), n/4 )

#x[1] = 0
#for (i in 2:n) x[i] = x[i-1] + s[i] + w[i]
#x = ts(x, start=0, frequency=4 )


x = scan("1_10.csv")
x = ts(x, start=0, frequency=4 )
plot(x, type="l")

q1 = ifelse(cycle(x)==1,1,0)
q2 = ifelse(cycle(x)==2,1,0)
q3 = ifelse(cycle(x)==3,1,0)
q4 = ifelse(cycle(x)==4,1,0)
A = ts.intersect( x, xL1=lag(x,-1), q1, q2, q3, q4, dframe=TRUE )

fit01 = lm( x ~ 0, offset=xL1, data=A )
fit02 = lm( x ~ 1, offset=xL1, data=A )

summary(fit01)

## 
## Call:
## lm(formula = x ~ 0, data = A, offset = xL1)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.8887 -0.8441  0.0438  0.8474  2.8330 
## 
## No Coefficients
## 
## Residual standard error: 1.183 on 499 degrees of freedom

summary(fit02)

## 
## Call:
## lm(formula = x ~ 1, data = A, offset = xL1)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.9054 -0.8608  0.0271  0.8307  2.8163 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)
## (Intercept)  0.01671    0.05299   0.315    0.753
## 
## Residual standard error: 1.184 on 498 degrees of freedom

AIC(fit01,fit02)

##       df      AIC
## fit01  1 1585.543
## fit02  2 1587.443

BIC(fit01,fit02)

##       df      BIC
## fit01  1 1589.756
## fit02  2 1595.869

The model with no drift is preferred according to both AIC and BIC.

r01 = residuals( fit01 )                                                                                       
plot(r01, type="l")                                                                                                                  

acf(r01, lag.max=50)

The ACF suggests quarterly seasonality.

fit03 = lm( x ~ 0 + q1 + q2 + q3 + q4, offset=xL1, data=A )
summary(fit03)

## 
## Call:
## lm(formula = x ~ 0 + q1 + q2 + q3 + q4, data = A, offset = xL1)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.3128 -0.7443  0.0240  0.7246  2.4770 
## 
## Coefficients:
##    Estimate Std. Error t value Pr(>|t|)    
## q1  0.42299    0.09236   4.580 5.89e-06 ***
## q2  0.68176    0.09199   7.411 5.45e-13 ***
## q3 -0.84716    0.09199  -9.209  < 2e-16 ***
## q4 -0.18749    0.09199  -2.038   0.0421 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.028 on 495 degrees of freedom
## Multiple R-squared:  0.9903, Adjusted R-squared:  0.9902 
## F-statistic: 1.265e+04 on 4 and 495 DF,  p-value: < 2.2e-16

The coefficient for Q1 is significant with a p-value of less than 10^{-5}. Reject null hypothesis. The joint hypothesis is also rejected with a p value of less than 10^{-15}.

r03 = residuals(fit03)
plot(r03,type="l")

acf(r03, lag.max=50)

AIC(fit01,fit02,fit03)

##       df      AIC
## fit01  1 1585.543
## fit02  2 1587.443
## fit03  5 1450.087

BIC(fit01,fit02,fit03)

##       df      BIC
## fit01  1 1589.756
## fit02  2 1595.869
## fit03  5 1471.150

Model 3 is strongly preferred by both AIC and BIC. Residuals look much more like white noise.