INTRODUCTION

The purpose of this assignment is to practice AR model and MA model by using Real Personal Consumption Expenditures.

IMPORTING DATA

library(Quandl)

y <- Quandl("FRED/PCECC96", type="zoo")
str(y)
## 'zooreg' series from 1947 Q1 to 2016 Q4
##   Data: num [1:280] 1199 1219 1223 1224 1230 ...
##   Index: Class 'yearqtr'  num [1:280] 1947 1947 1948 1948 1948 ...
##   Frequency: 4

DATA ANALYSIS

  1. Consider the quarterly Real Personal Consumption Expenditures, FRED/PCECC96.

A. I graph the time series data without transformation

plot(y, xlab="Years", ylab = "Real Personal Consumption Expenditures")

The graph has a steady increasing trend.

B. I construct the time series using:

\[ y_t = \Delta log_{c_t} = log_{c_t} - log_{c_{t-1}} \]
where \(c_{t}\) is the original quarterly Real Personal Consumption Expenditures.

* Plot the Data with transformation

dly <- diff(log(y))
plot(dly,xlab="Time", ylab="yt")

The graph is approximately stationary.

  1. Use ACF and PACF to identify suitable AR and/or MA model(s)
library(forecast)
par(mfrow=c(2,1), cex=1, mar=c(3,4,3,3))
Acf(dly, type='correlation', lag=36)
Acf(dly, type='partial', lag=36)



In the results, the second lag in ACF and the second and fourth lag in PACF are significant.

Therefore, I will examine the following
- AR(2) Model
- MA(2) Model
- MA(4) Model

  1. Check the estimated models (ACF, PACF, BIC, Ljung-Box Statistic and Maximum Likelihood, adequacy)

A. AR(2)

m1 <- Arima(dly, order=c(2,0,0))
m1
## Series: dly 
## ARIMA(2,0,0) with non-zero mean 
## 
## Coefficients:
##          ar1     ar2  intercept
##       0.0614  0.3185     0.0081
## s.e.  0.0566  0.0566     0.0007
## 
## sigma^2 estimated as 5.955e-05:  log likelihood=962.67
## AIC=-1917.34   AICc=-1917.2   BIC=-1902.82
tsdiag(m1, gof.lag=24)

plot.Arima(m1)

BIC(m1)
## [1] -1902.818

B. MA(2)

m2 <- Arima(dly, order=c(0,0,2))
m2
## Series: dly 
## ARIMA(0,0,2) with non-zero mean 
## 
## Coefficients:
##          ma1     ma2  intercept
##       0.0277  0.3652     0.0082
## s.e.  0.0564  0.0581     0.0006
## 
## sigma^2 estimated as 5.878e-05:  log likelihood=964.46
## AIC=-1920.92   AICc=-1920.78   BIC=-1906.4
tsdiag(m2, gof.lag=24)

plot.Arima(m2)

BIC(m2)
## [1] -1906.397

C. MA(4)

m3 <- Arima(dly, order=c(0,0,4))
m3
## Series: dly 
## ARIMA(0,0,4) with non-zero mean 
## 
## Coefficients:
##          ma1     ma2     ma3      ma4  intercept
##       0.0555  0.3671  0.0715  -0.0045     0.0082
## s.e.  0.0600  0.0605  0.0575   0.0742     0.0007
## 
## sigma^2 estimated as 5.887e-05:  log likelihood=965.23
## AIC=-1918.46   AICc=-1918.15   BIC=-1896.68
tsdiag(m3, gof.lag=24)

plot.Arima(m3)

BIC(m3)
## [1] -1896.676

Conclusion

1.Result

A. AIC: MA(2) has the smallest value(AIC=-1920.92). B. BIC: MA(2) has the smallest value(BIC=-1906.397). C. P-value for Ljung-Box statistic: MA(4) has many significant lag(lag 1 to lag 7) but MA(2) and AR(2) also have significant lags. D. Adequacy : In AR(2), MA(2), and MA(4) model, all roots are in the unit circle. Thus, all model are stationary and invertible

2. Conclusion

Therefore, MA(2) model is the best among models.