# The Box and Jenkins Method

This method fits Autoregressive Integrated Moving Average (ARIMA) type models. We have already considered simple AR and MA type models in Chapters 5 and 6. The Box-Jenkins methodology differences the series and looks at the sample correlogram of the differenced series to see whether stationarity is achieved. As will be clear shortly, if we have to difference the series once, twice or three times to make it stationary, this series is integrated of order 1, 2 or 3, respectively. Next, the Box-Jenkins method looks at the autocorrelation function and the partial autocorrelation function (synonymous with partial correlation coefficients) of the resulting stationary series to identify the order of the AR and MA process that is needed. The partial correlation between yt and yt-s is the correlation between those two variables holding constant the effect of all intervening lags, see Box and Jenkins (1970) for details. Figure 14.3 plots an AR(1) process of size T = 250 generated as yt = 0.7yt-i + et with et ~ IIN(0,4). Figure

14.3 shows that the correlogram of this AR(1) process declines geometrically as s increases. Similarly, Figure 14.5 plots an MA(1) process of size T = 250 generated as yt = et + 0.4et-1 with et ~ IIN(0,4). Figure 14.6 shows that the correlogram of this MA(1) process is zero after the first lag, see also problems 1 and 2 for further analysis. Identifying the right ARIMA model is not an exact science, but potential candidates emerge. These models are estimated using maximum likelihood techniques. Next, these models are subjected to some diagnostic checks. One commonly used check is to see whether the residuals are White noise. If they fail this test, these models are dropped from the list of viable candidates. Figure 14.3 AR(1) Process, p = 0.7

If the time-series is White noise, i. e., purely random with constant mean and variance and zero autocorrelation, then ps = 0 for s > 0. In fact, for a White noise series, if T ^<x>, VTps will be asymptotically distributed N(0,1). A joint test for Ho; ps = 0 for s = 1,2,…,m lags, is given

 ★★★★★★ ★★★★★★ 1 0.725 0.725 132.99 0 ★★★★ 2 0.503 -0.048 197.27 0 ★★★ 3 0.33 -0.037 225.05 0 ★★ 4 0.206 -0.016 235.92 0 ★ 5 0.115 -0.022 239.33 0 • 1 • 6 0.036 -0.05 239.67 0 • 1 • 7 -0.007 0.004 239.68 0 • 1 • 8 -0.003 0.05 239.68 0 • 1 • 9 -0.017 -0.041 239.75 0 ★ 10 -0.06 -0.083 240.71 0 ★ 11 -0.11 -0.063 243.91 0 ★ 12 -0.04 0.191 244.32 0
 Autocorrelation Partial Correlation AC PAC Q-Stat Prob Figure 14.4 Correlogram of AR(1)

by the Box and Pierce (1970) statistic (14.1)

This is asymptotically distributed under the null as Xm,. A refinement of the Box-Pierce Q – statistic that performs better, i. e., have more power in small samples is the Ljung and Box (1978) Qlb statistic

 Autocorrelation Partial Correlation AC PAC Q-Stat Prob ~k~k~k 1 0.399 0.399 40.240 0.000 ~k 2 0.033 -0.150 40.520 0.000 ~k 3 0.010 0.066 40.545 0.000 4 -0.002 -0.033 40.547 0.000 ~k ~k 5 0.090 0.127 42.624 0.000 6 0.055 -0.045 43.417 0.000 7 0.028 0.042 43.625 0.000 ~k 8 -0.031 -0.075 43.881 0.000 9 -0.034 0.023 44.190 0.000 10 -0.027 -0.045 44.374 0.000 11 -0.013 0.020 44.421 0.000 ~k ~k 12 0.082 0.086 46.190 0.000
 Figure 14.6 Correlogram of MA(1)

Qlb = T(T + 2) Em=i?2/(T – j) (14.2)

This is also distributed asymptotically as xL under the null hypothesis. Maddala (1992, p. 540) warns about the inappropriateness of the Q and Qlb statistics for autoregressive models. The arguments against their use are the same as those for not using the Durbin-Watson statistic in autoregressive models. Maddala (1992) suggests the use of LM statistics of the type proposed by Godfrey (1979) to evaluate the adequacies of the ARMA model proposed.

For the consumption series, T = 49 and the 95% confidence interval for ~ра is 0 ± 1.96 (1/%49) which is ±0.28. Figure 14.2 plots this 95% confidence interval as two solid lines around zero. It is clear that the sample correlogram declines slowly as the number of lags s increase. Moreover, the Qlb statistics which are reported for lags 1, 2, up to 13 are all statistically significant. These were computed using EViews. Based on the sample correlogram and the Ljung-Box statistic, the consumption series is not purely random white noise. Figure 14.7 plots the sample correlogram for ACt = Ct — Ct-1. Note that this sample correlogram dies out abruptly after the first lag. Also, the Qlb statistics are not significant after the first lag. This indicates stationarity of the first-differenced consumption series. Problem 3 asks the reader to plot the sample correlogram for personal disposable income and its first difference, and to compute the Ljung-Box Qlb statistic to test for purely White noise based on 13 lags.

A difficult question when modeling economic behavior is to decide on what lags should be in the ARIMA model, or the dynamic regression model. Granger et al. (1995) argue that there are disadvantages in using hypothesis testing to help make model specification decisions based on the data. They recommend instead the use of model selection criteria to make those decisions.

The Box-Jenkins methodology has been popular primarily among forecasters who claimed better performance than simultaneous equations models based upon economic theory. Box – Jenkins models are general enough to allow for nonstationarity and can handle seasonality. However, the Box-Jenkins models suffer from the fact that they are devoid of economic theory and as such they are not designed to test economic hypothesis, or provide estimates of key elasticity parameters. As a consequence, this method cannot be used for simulating the effects

Autocorrelation Partial Correlation AC PAC Q-Stat Prob

 1 0.465 0.465 11.059 0.001 2 0.065 -0.194 11.277 0.004 ~k ~k 3 -0.129 -0.099 12.16 0.007 ~k 4 -0.048 0.101 12.288 0.015 5 -0.012 -0.053 12.296 0.031 6 0.015 0.015 12.309 0.055 7 -0.016 -0.027 12.324 0.09 ~k ~k 8 -0.115 -0.133 13.123 0.108 ~k 9 -0.081 0.056 13.528 0.14 ~k ~k 10 0.124 0.194 14.501 0.151 ~k 11 0.194 0.001 16.944 0.11 ~k ~k 12 0.098 -0.027 17.589 0.129 ~k 13 0.063 0.12 17.861 0.163 14 0.034 -0.017 17.941 0.209 15 0.027 0.016 17.994 0.263 16 0.028 0.034 18.051 0.321 17 0.026 -0.026 18.105 0.382 ~k ~k~k 18 -0.152 -0.193 19.959 0.335 ~k~k 19 -0.029 0.279 20.027 0.393 ~k 20 0.101 0.016 20.902 0.403

Figure 14.7 Correlogram of First Difference of Consumption

of a tax change or a Federal Reserve policy change. One lesson that economists learned from the Box-Jenkins methodology is that they have to take a hard look at the time-series properties of their variables and properly specify the dynamics of their economic models. Another popular forecasting technique in economics is the Vector Autoregression (VAR) methodology proposed by Sims (1980). This will be briefly discussed next.