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.

image603

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

0.000

★★★★

2

0.503

-0.048

197.27

0.000

★★★

3

0.330

-0.037

225.05

0.000

★★

4

0.206

-0.016

235.92

0.000

5

0.115

-0.022

239.33

0.000

• 1 •

6

0.036

-0.050

239.67

0.000

• 1 •

7

-0.007

0.004

239.68

0.000

• 1 •

8

-0.003

0.050

239.68

0.000

• 1 •

9

-0.017

-0.041

239.75

0.000

10

-0.060

-0.083

240.71

0.000

11

-0.110

-0.063

243.91

0.000

12

-0.040

0.191

244.32

0.000

Autocorrelation Partial Correlation AC PAC Q-Stat Prob

Подпись:
Figure 14.4 Correlogram of AR(1)

by the Box and Pierce (1970) statistic

Подпись: Q(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

12.160

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

0.090

~k

~k

8

-0.115

-0.133

13.123

0.108

~k

9

-0.081

0.056

13.528

0.140

~k

~k

10

0.124

0.194

14.501

0.151

~k

11

0.194

0.001

16.944

0.110

~k ~k

12

0.098

-0.027

17.589

0.129

~k

13

0.063

0.120

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.

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