Econometrics Using STATA: Box Jenkins Approach To Building ARMA Models (Statistics Project Sample)

CPI ECONOMETRICS MODELLING
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1 Box-Jenkins approach to building ARMA models
Box and Jenkins popularized the approach of using the moving average and the autoregressive model (Baum, 2006). Although in the beginning, the autoregressive and moving average models were already known, Box and Jenkins, introduced a systematic idea for estimating models that incorporates both approaches (Perez-Truglia, 2009). Therefore, the Box-Jenkins ARMA model is a combination of the Autoregressive model and the Moving Average model as shown below:
yt=a0+a1yt-1+a2yt-2…apyt-p-b1ut-1-b2ut-2-…-bqut-q+ut
There are three-core stages in building a Box-Jenkins time series model: model identification, model estimation, and model validation (Cameron and Trivedi, 2010). The class of ARMA models is quite robust; therefore, a researcher must decide which model is the most appropriate during the model identification phase. Model estimation is fitting the model using the least squares method and finally, the model is validated by carrying out a normality test of the residues to ensure there is no white noise (Moody, 2009).
2 Calculate the logarithmic change of the series
∆cpit=cpit-cpit-1
Where cpit=ln⁡(CPIt)
* In a stationary series, statistical properties such as the mean, variance, and autocorrelation remain constant over time. Most statistical models are based on the assumption that all the variables can be stationarized (Baum, 2006). Figure 1 below is a graph of the natural log of the consumer price index. As it can be seen, the data at this point is not stationary since it does not create a bell shaped (Perez-Truglia, 2009). Figure 2 is a graph of the first difference of the log, which is normally shaped, implying the data is stationary.
Figure SEQ Figure \* ARABIC 1: Graph for the log of cpi
Figure SEQ Figure \* ARABIC 2a: Histogram for the logarithmic difference of cpi
Figure 2b: Time series graph for log difference of CPI
* The auto-correlation function (ACF) of a stationary process at lag h; ρxh=Corr(Xt,Xt-h) measures the linear dependency between the process variables Xt and Xt-h. However, the intermediate variables Xs, where t-h
Process

ACF

PACF

White noise

Statistically insignificant coefficients

Statistically insignificant coefficients

AR(2)

Declines gradually based on a geometric equation

The first 2 coefficients are significant, all others are insignificant

MA(1)

The first ACF are significant while all others are insignificant

Slowly declining graph or sinusoid

ARMA(2,1)

Slowly decaying graph ACF

Geometrically decreasing graph PACF

* Exploring the Autocorrelation function and partial autocorrelation graphs revels two key points. The Autocorrelation function is declining geometrically though does not behave very well after the eighth lag and the partial autocorrelation function has a sharp cut off after three lags (Moody, 2009). This might suggest a good Box-Jenkins model process to use in this case is an autoregressive model or order 3 AR (3). Figure 3 is a graph for the autocorrelation function for the variable ∆cpit. According to the graph, the ACF dies off slowly and since the ACF is significantly different from zero by lag (12), then we can conclude that there is no white noise (Burke, 2009).
Figure SEQ Figure \* ARABIC 3: ACF
Figure 4 shows the trend in the PACF. The data shows that the PACF is non-zero for the first few lags that is when considering lags before 12. Again, this suggests that an Autoregressive model would be the best fit for this data (Wooldridge, 2015).
Figure SEQ Figure \* ARABIC 4: PACF
* Table 1 below shows the approximation for ∆cpit over the period 1960Q1 to 2009Q4. Examining the data shows that the AR coefficients are significant from order (0, 0) to (3, 3) at the 0.005 significance level. Looking at the MA coefficients shows they are significant except for ARMA (1,2), ARMA (3,2) and ARMA (3,3) at the 0.005 significance level. The Wald test is used to determine whether explanatory variables in a model are significant (Burke, 2009). Table (1) shows the values for the Wald test and their significance level. According to the data, the variables are significant at the 0.005 level, p-value=0. Finally, the likelihood ratio test is used to evaluate the difference between nested models (Chen, 2010). These values can be used to test what happens when two models are nested together.
Table SEQ Table \* ARABIC 1: Model selection
ARMA (p,q)

 

Coefficients

P-value

Log-likelihood

Wald-Chisq

AIC

BIC

AR(1)

ar

0.754

0.000

769.6936

380.04

-1533.387

-1523.507

AR(2)

ar

0.649

0.000

739.9211

313.9

-1473.842

-1463.962

AR(3)

ar

0.663

0.000

743.4247

387.87

-1480.849

-1470.969

MA(1)

ma

0.663

0.000

740.4898

145.03

-1474.98

-1465.1

MA(2)

ma

0.420

0.000

717.7038

89.23

-1429.408

-1419.528

MA(3)

ma

0.538

0.000

723.7384

110.8

-1441.477

-1431.597

ARMA(1,1)

ar

0.914

0.000

777.1267

1122.4

-1546.253

-1533.08


ma

-0.421

0.000





ARMA(1,2)

ar

0.837

0.000

772.4422

668.32

-1536.884

-1523.711


ma

-0.212

0.030





ARMA(1,3)

ar

0.687

0.000

777.7114

321.45

-1547.423

-1534.25


ma

0.275

0.000





ARMA(2,1)

ar

0.552

0.000

770.1007

286.08

-1532.201

-1519.028


ma

0.789

0.000





ARMA(2,2)

ar

0.866

0.000

747.5864

535.33

-1487.173

-1474


ma

-0.400

0.000





ARMA(2,3)

ar

0.537

0.000

744.9912

266.14

-1481.982

-1468.809


ma

0.244

0.000





ARMA(3,1)

ar

0.553

0.000

769.2345

548.42

-1530.469

-1517.296


ma

0.450

0.000





ARMA(3,2)

ar

0.570

0.000

746.983

395.27

-1485.966

-1472.793


ma

0.219

0.031





ARMA(3,3)

ar

0.751

0.000

744.4003

372.02

-1480.801

-1467.627

 

ma

-0.157

0.070

 

 



All the coefficients were significant at the 0.05 levels expect for ARMA (3, 3) where the moving average coefficient is insignificant with a p-value of 0.07. AIC and SBIC are flexible measures used to determine which model is of best fit (Asteriou and Hall, 2015). In order to choose a model based on the AIC and SBIC measure, we choose the model with the lowest value. Based on these values, the prefe...