Econometrics Using STATA Box Jenkins Approach To Building ARMA Models - 1100 Words

Econometrics Using STATA: Box Jenkins Approach To Building ARMA Models (Statistics Project Sample) Content: CPI ECONOMETRICS MODELLINGBy 1 Box-Jenkins approach to building ARMA modelsBox 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-2apyt-p-b1ut-1-b2ut-2--bqut-q+utThere 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-1Where 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 cpiFigure SEQ Figure \* ARABIC 2a: Histogram for the logarithmic difference of cpiFigure 2b: Time series graph for log difference of CPI * The auto-correlation function (ACF) of a stat ionary process at lag h; xh=Corr(Xt,Xt-h) measures the linear dependency between the process variables Xt and Xt-h. However, the intermediate variables Xs, where t-hProcess 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: ACFFigure 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 explanator y 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 selectionARMA (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...