Box-Jenkins Approach to ARMA Models & Analysis of US CPI Data

This article explains the Box-Jenkins approach to building ARMA models and analyzes the US Consumer Price Index (CPI) data from 1960Q1 to 2012Q4. It covers the identification, estimation, and diagnostic checking steps of the Box-Jenkins method, as well as ACF and PACF. The article also discusses the significance of ACF coefficients and identifies the preferred model order based on AIC and SBIC criteria. Finally, it estimates the ARIMA model and reports the results.

1. Explain the Box-Jenkins approach to building ARMA models.
This method has been developed by Box and Jenkins to identify and estimate the model for specific
time series, which is called as Box-Jenkins method. Auto regressive integrated moving average
(ARIMA) model is known as autoregressive moving average (ARMA) is particularly used in
econometrics and time series analysis for forecasting the future Box et.al (1974), Box et.al (1973) and
Ljung and Box (1978). The method is well fitted to long and medium time series at least 50
observations. We call times time series has a sequenced though time. The sequence can be denoted by
X1, X2, X3…. Xt , where t is known as time period and X are variables. If X is determined by
mathematical rule we call it deterministic, if only future value can be predicted by probability we call
it statistical. Statistical process is known as stationary stochastic process Cramér and Leadbetter
(2013). If distribution is same for all values of t then the statistical process is stationary, which means
mean and variance is constant for all values of t. A series having trend is non-stationary because it
depends on value of t. It is important in Box-Jenkins method to make non-stationary data into a
stationary data. To check the statinarity we can apply Dickey and Fuller (1981) test of unit root.
The Box-Jenkins approach is based on iteration approach that consists of the following 3 steps.
1. Identification: we used it for plotting of the data, autocorrelation and partial correlation that
best summarizes the data.
In identification we further divide it into two parts.
Differencing: We can difference the data on the basis of unit root test, by which we can make non-
stationary data, stationary data by first or second order differencing.
Configuring AR and MA
Auto correlation function: The first step summarizes the autocorrelation and X-axis shows the lag
and the y-axis gives correlation coefficient between -1 to 1.
Partial autocorrelation function: The correlation is collected for the lag values that are not
accounted.
2. Estimation: The estimation of Box-Jenkins can be done by Maximum Likelihood technique.
This estimation is done for checking the autocorrelation of the residuals. If the autocorrelation
is small the model is fine and forecast can be generated. If the autocorrelation is large then p
and q are adjusted and re-estimated accordingly.
3. Diagnostic Checking: we evaluate the model on the bases of available data by taking
autocorrelation into consideration. Weather the model is good fit or not.
Over fitting: If the model is complex, that means the model over fits the data, we need to calculate
random noise. This problem arises in time series forecasting analysis because of negative results for
poor forecasting.
Residual errors:
By calculating residuals errors can be reduced, which means errors in model would resemble white
noise, which is Gaussian distribution with zero mean and uniform variance. We can use plots,

histograms and Q-Q plots. Otherwise we can check it by ACF and PACF plot of residual error time
series.
2. Open the workfile "cpi.dta". The file contains quarterly data of the U.S. Consumer Price Index
(CPI) over the period 1960Q1 to 2012 Q4. Calculate the logarithmic change of the series, i.e.:
𝛥𝑐𝑝𝑖𝑡=𝑐𝑝𝑖𝑡−𝑐𝑝𝑖𝑡−1
where 𝑐𝑝𝑖𝑡=ln(𝐶𝑃𝐼𝑡) (Note: Use the natural logarithm).
a) Use the full sample period to obtain a graph of the Δ𝑐𝑝𝑖 series. Based on the graph, does the
series appear to be stationary? Comment on the behavior of the series over time.
The logarithmic change of data for U.S. Consumer Price Index (CPI) over the period 1960Q1 to 2012
Q4 has been calculated by taking Log of variable CPI.
3.2
3.6
4.0
4.4
4.8
5.2
5.6
60 65 70 75 80 85 90 95 00 05 10
CPI1
Source: Authors Competitions
Note: The variable CPI seems stationary there is no fluctuation over the period of time.
The graph of CPI over the time period 1960Q1 to 2012 Q4 has been calculated. The graph is based on
log variables, based on graph it look the variable is stationary for all period of time. It means the data
have constant mean, variance and autocorrelation, etc. are all constant over time.
b) Read Chapter 6 in Brooks (2014) and briefly explain the terms autocorrelation function (ACF) and
partial autocorrelation function (PACF). What shape would these two functions take for a stationary
autoregressive process, a moving average process, and a combination autoregressive moving average
process?
Autocorrelation function (ACF) is based on AR (1) model which predicts the present value of a times
series by taking immediate values of time Wu and Wei (1989). ACF means series is weakly stationary
series. Which means autocorrelation for any participant lag is same. It means mean, variance,
covariance and correlation between Xt is same for all t.
Partial Autocorrelation function (PACF) is known as conditional correlation. In which we consider two
variables under the assumption by taking value of other set of variables.

We can consider a regression equation in which y is response variable x1, x2, x3 are predictive
variables. Partial correlation between y and x3 is correlation between determined by y and x3 are related
to x1 and x2. Partial correlation can be done by correlating residuals. One is predicting from y to x1 and
x2and second from x3 to x1 and x2. We have done partial correlation from y to x3 predicted by x1 and x2.
c) Obtain the ACF and PACF for Δ𝑐𝑝𝑖 in Stata (specify the number of autocorrelations to be 12)
using data from 1960Q1 to 2009Q4 (Note that this is not the full sample). Discuss the significance of
the ACF coefficients. Based on your answer to Question 2(b), try to identify the tentative models that
you would want to estimate.
Table1: Results of ACF and PACF
Autocorrelation Partial Correlation AC PAC Q-Stat Prob
.|*** | .|*** | 1 0.461 0.461 45.499 0.000
.|** | .|. | 2 0.241 0.036 57.989 0.000
.|** | .|* | 3 0.270 0.186 73.769 0.000
.|* | .|. | 4 0.199 0.008 82.389 0.000
.|** | .|** | 5 0.298 0.232 101.79 0.000
.|** | .|* | 6 0.304 0.078 122.11 0.000
.|* | .|. | 7 0.194 -0.006 130.40 0.000
.|* | .|. | 8 0.207 0.062 139.85 0.000
.|* | *|. | 9 0.113 -0.091 142.68 0.000
.|* | .|. | 10 0.107 0.026 145.25 0.000
.|* | .|. | 11 0.157 0.013 150.81 0.000
.|* | .|* | 12 0.180 0.088 158.11 0.000
The results if ACF shows that there is no trend in the data, because of fluctuation from 0.199 at lag 4
to 0.304 at lag 6. We can estimate simple liner regression model and if we can include some more
variables in our study we can go for short-run and long-run dynamics and causality analysis. The study
has a short coming that we are using one a single variables for our study. We need to include more
independent variables to find exact relationship.
d) Estimate all ARMA models from order (0,0) to (3,3) for Δ𝑐𝑝𝑖 over the sample period 1960Q1 to
2009Q4. From your estimations which is the preferred model order? Explain why? (You do not need to
report coefficient estimates for all the estimated models; just report the value of the AIC and SBIC
criteria in a Table).
We have estimated the model on the basis of Akaike information criterion (AIC) and Schwarz criterion
(SBIC) information criteria. For the estimation of model we need to find out AIC and SBIC lag length
Teräsvirta and Mellin (1986) and Burnham and Anderson (2004). We first take the lag order 0,0 and
the 3,3. The table below gives the results of information criteria. Based on the below table 2, we can
choose (2,3) (0,0) is a best model because the lower the value of Akaike of information criteria the
better the model. The value of AIC (2.178715) is lowest. In table 2 there is a comparison when we take
lag order 0,0 and 3,3 which model is best, based on the results model 3,3 is best because of AIC and
SBIC value is lower.
Table 1: AIC and SBIC based on lag order (0,0)
Model AIC BIC HQ
(0,0) (0,0) 2.573096 2.604973 2.585983

Table 2: AIC and SBIC based on lag order (0,0)
Model AIC BIC HQ
(2,3)(0,0) 2.1787 2.2902 2.2238
(3,3)(0,0) 2.1863 2.3138 2.2378
(0,2)(0,0) 2.2183 2.2821 2.2441
(1,2)(0,0) 2.2217 2.3014 2.2539
(0,3)(0,0) 2.2295 2.3041 2.2566
(2,2)(0,0) 2.2295 2.3251 2.2681
(1,3)(0,0) 2.2303 2.3259 2.2689
(1,1)(0,0) 2.2315 2.2952 2.2572
(2,1)(0,0) 2.2344 2.3141 2.2666
(3,2)(0,0) 2.2353 2.3468 2.2804
(3,1)(0,0) 2.2367 2.3323 2.2753
(0,1)(0,0) 2.2877 2.3355 2.3070
(3,0)(0,0) 2.3620 2.4417 2.3942
(2,0)(0,0) 2.3697 2.4334 2.3954
(1,0)(0,0) 2.4901 2.5379 2.5094
(0,0)(0,0) 2.5730 2.6049 2.5859
e) Re-estimate the preferred model(s) from Question 2(d). Again, use only the sample 1960Q1 to
2009Q4. Report and comment on the results.
By estimating the ARIMA model over the sample period of 1960Q1 to 2009Q4, the results are given
below in table 3.
Table 3: AIC and SBIC based on lag order (0,0)
Model AIC BIC HQ
(0,0) (0,0) 2.589861 2.623076 2.603305
Table 4: AIC and SBIC based on lag order (0,0)
Model AIC BIC HQ
(2,3)(0,0) 2.1674 2.2837 2.2145
(3,3)(0,0) 2.1898 2.3227 2.2436
(0,2)(0,0) 2.2301 2.2965 2.2570
(1,2)(0,0) 2.2315 2.3145 2.2651
(0,3)(0,0) 2.2356 2.3187 2.2693
(1,1)(0,0) 2.2405 2.3069 2.2674
(2,2)(0,0) 2.2406 2.3402 2.2809
(1,3)(0,0) 2.2410 2.2407 2.2814
(3,1)(0,0) 2.2426 2.3422 2.2825
(3,2)(0,0) 2.2435 2.3597 2.2909
(2,1)(0,0) 2.2464 2.3295 2.2800
(0,1)(0,0) 2.2831 2.3329 2.3032
(3,0)(0,0) 2.3559 2.4389 2.3895
(2,0)(0,0) 2.3581 2.4245 2.3850
(1,0)(0,0) 2.4919 2.5417 2.5120
(0,0)(0,0) 2.5898 2.6230 2.6033

We do not find much changes in the results, the best model out of (0,0) and (3,3), is (3,3) model
because the AIC (2.1898) value is lowest.
f) Perform diagnostic checks on the residuals from the estimated model(s) in Question 2(e). Does the
model fit the data well?
Figure 1: Residual Graph from diagnostic testing
-10
-8
-6
-4
-2
0
2
4
1 / 1 / 6 0
1 / 1 / 6 2
1 / 1 / 6 4
1 / 1 / 6 6
1 / 1 / 6 8
1 / 1 / 7 0
1 / 1 / 7 2
1 / 1 / 7 4
1 / 1 / 7 6
1 / 1 / 7 8
1 / 1 / 8 0
1 / 1 / 8 2
1 / 1 / 8 4
1 / 1 / 8 6
1 / 1 / 8 8
1 / 1 / 9 0
1 / 1 / 9 2
1 / 1 / 9 4
1 / 1 / 9 6
1 / 1 / 9 8
1 / 1 / 0 0
1 / 1 / 0 2
1 / 1 / 0 4
1 / 1 / 0 6
1 / 1 / 0 8
RESID
The residual of are summed to zero and have a mean to zero. We will see Radom scattering of point,
nothing gives any indication the assumption of model are false. The variability is same in all point of
time and no Random curvature in the model. The model is normal by seeing the plot of residuals. The
model can be used for analysis. The variance of model is constant which is moving along the line zero.
g) Use the model estimated in Question 2(e) to generate one step ahead (static) forecasts in Stata for
the period 2010Q1 – 2012Q4. Create a graph of the actual Δ𝑐𝑝𝑖 series and the forecasts that you have
generated over the specified out-of-sample period. Comment on the results.
Figure 1 gives the static graph of forecasts, the blue line is the actual variable CPI and forecasted
values are red dotted. The basic criteria are the blue lie should pass through the line with without
passing them, which means our model is best. The value of root mean square is more than 50 percent
which means the error in actual CPI and forecasted CPI, they are not moving close to each other which
means, so the regression model is not good fitted.
Figure 1: Forecasted graph of CPI from 2009 Q1 to 2012 Q4.

210
215
220
225
230
235
CPIF ± 2 S.E.
Forecast: CPIF
Actual: CPI
Forecast sample: 1/01/2009 1/10/2012
Included observations: 16
Root Mean Squared Error 0.788292
Mean Absolute Error 0.641514
Mean Abs. Percent Error 0.290597
Theil Inequality Coefficient 0.001776
Bias Proportion 0.001320
Variance Proportion 0.068431
Covariance Proportion 0.930249
Theil U2 Coefficient 0.569055
Symmetric MAPE 0.290625
References
Box, G. E., Jenkins, G. M., & MacGregor, J. F. (1974). Some recent advances in forecasting and
control. Applied Statistics, 158-179.
Box, G. E. P., & Jenkins, G. M. (1973). Some comments on a paper by Chatfield and Prothero and on
a review by Kendall. Journal of the Royal Statistical Society. Series A (General), 136(3), 337-352.
Ljung, G. M., & Box, G. E. (1978). On a measure of lack of fit in time series
models. Biometrika, 65(2), 297-303.
Cramér, H., & Leadbetter, M. R. (2013). Stationary and related stochastic processes: Sample function
properties and their applications. Courier Corporation.
Dickey, D. A., & Fuller, W. A. (1981). Likelihood ratio statistics for autoregressive time series with a
unit root. Econometrica: Journal of the Econometric Society, 1057-1072.
Wu, J. P., & Wei, S. (1989). Time series analysis. Hunan Science and Technology Press, ChangSha.
Teräsvirta, T., & Mellin, I. (1986). Model selection criteria and model selection tests in regression
models. Scandinavian Journal of Statistics, 159-171.
Burnham, K. P., & Anderson, D. R. (2004). Multimodel inference: understanding AIC and BIC in
model selection. Sociological methods & research, 33(2), 261-304.