# Time-Series Analysis

There has been an enormous amount of research in time-series econometrics, and many economics departments have required a time-series econometrics course in their graduate sequence. Obviously, one chapter on this topic will not do it justice. Therefore, this chapter will focus on some of the basic concepts needed for such a course. Section 14.2 defines what is meant by a stationary time-series, while sections 14.3 and 14.4 briefly review the Box-Jenkins and Vector Autoregression (VAR) methods for time-series analysis. Section 14.5 considers a random walk model and various tests for the existence of a unit root. Section 14.6 studies spurious regressions and trend stationary versus difference stationary models. Section 14.7 gives a simple explanation of the concept of cointegration and illustrates it with an economic example. Finally, section 14.8 looks at Autoregressive Conditionally Heteroskedastic (ARCH) time-series.

Figure 14.1 plots the consumption and personal disposable income data considered in Chapter 5. This was done using EViews. This is annual data from 1959 to 2007 expressed in real terms. Both series seem to be trending upwards over time. This may be an indication that these time-series are non-stationary. Having a time-series xt that is trending upwards over time may invalidate all the standard asymptotic theory we have been relying upon in this book. In fact, J2t=i Xt/T may not tend to a finite limit as T and using regressors such as xt means that X’X/T does not tend in probability limits to a finite positive definite matrix, see problem 6.

Figure 14.1 U. S. Consumption and Income, 1959-2007 |

Non-standard asymptotic theory will have to be applied which is beyond the scope of this book, see problem 8.

Definition: A time-series process xt is said to be covariance stationary (or weakly stationary) if its mean and variance are constant and independent of time and the covariances given by cov(xt, xt-s) = Ys depend only upon the distance between the two time periods, but not the time periods per se.

In order to check the time-series for weak stationarity one can compute its autocorrelation function. This is given by ps= correlation (xt, xt-s) = Ys/Yo. These are correlation coefficients taking values between —1 and +1.

The sample counterparts of the variance and covariances are given by 7o = E Li(xt — x)2/T

Ys = E T=1s(xt — x)(xt+s — x)/T

and the sample autocorrelation function is given by 7s = Ys/Yo- Figure 14.2 plots 7s against s for the consumption series. This is called the sample correlogram. For a stationary process, ps declines sharply as the number of lags s increase. This is not necessarily the case for a nonstationary series. In the next section, we briefly review a popular method for the analysis of time-series known as the Box and Jenkins (1970) technique. This method utilizes the sample autocorrelation function to establish whether a series is stationary or not.

k’k’k’k’k’k’k |
‘k’k’k’k’k’k’k |
1 |
0.935 |
0.935 |
45.500 |
0.000 |
||

kkkkkk |
2 |
0.868 |
-0.050 |
85.527 |
0.000 |
|||

kkkkkk |
3 |
0.800 |
-0.042 |
120.26 |
0.000 |
|||

kkkkk |
4 |
0.733 |
-0.029 |
150.08 |
0.000 |
|||

kkkkk |
5 |
0.668 |
-0.024 |
175.39 |
0.000 |
|||

kkkk |
6 |
0.604 |
-0.030 |
196.57 |
0.000 |
|||

kkkk |
7 |
0.541 |
-0.029 |
214.00 |
0.000 |
|||

kkk |
8 |
0.480 |
-0.033 |
228.02 |
0.000 |
|||

kkk |
9 |
0.421 |
-0.015 |
239.12 |
0.000 |
|||

kkk |
10 |
0.369 |
0.004 |
247.83 |
0.000 |
|||

kk |
11 |
0.320 |
-0.009 |
254.57 |
0.000 |
|||

kk |
12 |
0.272 |
-0.033 |
259.57 |
0.000 |
|||

kk |
13 |
0.226 |
-0.027 |
263.10 |
0.000 |
|||

k |
14 |
0.181 |
-0.021 |
265.45 |
0.000 |
|||

~k |
15 |
0.140 |
-0.013 |
266.88 |
0.000 |
|||

~k |
16 |
0.097 |
-0.052 |
267.60 |
0.000 |
|||

17 |
0.055 |
-0.036 |
267.83 |
0.000 |
||||

18 |
0.011 |
-0.052 |
267.84 |
0.000 |
||||

19 |
-0.032 |
-0.034 |
267.93 |
0.000 |
||||

20 |
-0.073 |
-0.026 |
268.39 |
0.000 |

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